Short answer
I think “running” in “a running experiment” is the same part of speech as “running” in “an experiment that is running” (so to that extent, I agree with Greg Lee’s answer).
What to call this part of speech depends a bit on theory, but I think a fairly safe noncommittal term is “participle”. This is the “present” or “progressive” participle, of course, not the “past” or “passive” participle.
In mainstream analyses of English that I am familiar with (like CGEL 2002) , participles are classified as verb forms, so the part of speech would be “verb”.
However, there are also arguments for classifying participles as a type of “adjective”.
I would only call it a “gerund” if I was using an analysis that doesn’t distinguish between present participles and gerunds, and I think the usual terminology used in analyses like this is “gerund-participle”. (As far as I know, this unified analysis only occurs in theories which say the “gerund-participle”’s part of speech is “verb”.)
Long answer
This is a very difficult and deep question. As far as I can tell, not even linguists agree about what part of speech “running” is in a phrase like this, or how it functions grammatically. You have already brought up most of the relevant established facts; any explanation beyond that is likely to be debatable and dependent on specific linguistic theories. I know very little about these, but I did read a couple of papers that I found interesting and that might be of interest to you as well.
Terminology
Terminology is difficult, especially when dealing with different theoretical backgrounds. However, I think it might help to go over some terms and how they are commonly used to avoid confusion later on.
Participle. For people who use this term, it refers to certain word forms that in English are morphologically composed of a verb base and a participle-forming suffix. English has two types of participles.
The “past participle” is usually formed with the suffix -ed, or more rarely -en, -t, a zero suffix, or a vowel change. It appears in the passive construction after a form of to be (e.g. The pie was eaten by your cousin), and in the perfect construction after a form of to have (e.g. Your cousin has eaten the pie). Of course, not all word forms that look like participles are participles. The word form “kicked” looks like a past participle, but in a sentence like Pat kicked Sandy, “kicked” is not a past participle, it’s a past-tense verb. (But “kicked” is a past participle in Sandy was kicked by Pat.)
The “present participle” is always formed with the suffix -ing. It appears in the progressive construction after a form of to be (e.g. Your cousin is eating the pie). I think most linguists would say that not all words that look like present participles are present participles, but it’s hard for me to find a good theory-neutral example of a word that looks like a present participle but definitely isn’t. I guess the noun “gelding” works, as in The horse was a gelding: this looks like a present participle, but it definitely isn’t one. (In contrast, “gelding” would definitely be a present participle in a sentence like The farmer is gelding the horse.)
Gerund. For people who use this term, it refers to a certain word form that in English is morphologically composed of a verb base and a suffix -ing. It seems to be used to refer to slightly different sets of things by different people, but the common point is that gerunds are, or at least can be, “nouny”. I don’t think “running” in “a running experiment” bears much if any resemblance to a noun, so fortunately I think I can avoid discussing gerunds in the rest of this post.
Gerund-participle. It is a currently unsettled issue whether the “present participle” in English should be considered a distinct word form from the “gerund”. In standard English, there is in principle no way of distinguishing the two by either sound or spelling. That is why they are sometimes referred to as a single form called the “gerund-participle”; Pullum and Huddleston’s Cambridge Grammar of the English Language argues for this analysis.
Adjective. The meaning of “adjective” is fairly clear in one respect: it’s easy to give examples of words that anyone would agree are adjectives, such as happy, sad, important, colorless, historical, perfect, dead. It’s much more difficult to define exactly what makes these words adjectives; this is where you get competition between different theories. Linguists generally agree that, if a word behaves similarly in grammar to other words that are known to be adjectives, that constitutes evidence for it being an adjective as well.
Verb. In the analyses of English grammar that I have any familiarity with, verbs are defined as a class of words. In English, these words take different forms; for example, the verb steal takes the suffix -s in its third-person singular present-tense form “steals”. This is called “inflection”, and it isn’t considered to change steal into another word or another part of speech: it’s still the same word, and it’s still a verb. “Steals” is what is called a “finite” form of the verb: it can be used in an independent clause like The robber steals the treasure. The grammars I am familiar with also recognize “non-finite” inflected forms of verbs, namely, the infinitive (which is just “steal”; it doesn’t have a special suffix), the past participle “stolen”, and the present participle and gerund/gerund-participle “stealing”. These non-finite forms are also considered to be the same word as the finite forms of steal and the same part of speech; i.e. in Your cousin is eating the pie, “eating” is considered to be a verb: it is the word eat inflected into a certain form. Apparently, there are other approaches to dealing with the part-of-speech classification of non-finite forms, but I am not familiar with them. It seems in Distributive Morphology, a non-Lexicalist theory of grammar, “the same Vocabulary item may appear in different morphological categories depending on the syntactic context that the item’s l-morpheme (or Root) appears in”, so there are no such things as nouns, verbs and adjectives in the traditional sense. (“Distributed Morphology: Frequently Asked Questions List”, maintained by Rolf Noyer. Note that this is not a mainstream linguistic theory. Also, here is a related question on Linguistics SE: Are there any languages with minimal distinctions between the noun and verb categories?)
Participial adjective. As far as I can tell, people just use this as a term meaning “an adjective derived from a participle”. In other words, “participial adjective” is a label for a word that IS considered to be an adjective, and is NOT considered to be a participle. Hopefully I’ve understood this correctly. I can imagine some people might use it with the sense “an adjective that is simultaneously a participle”, but that isn’t what I would expect it to mean without further clarification.
Relevant points re-summarized
You already pointed these things out in your question, but I think it will help me organize my answer to repeat them. Also, these are the least controversial parts of this post.
At least some words that end in -ing are adjectives, like enlightening
As far as I know, everyone agrees that at least some words that end in -ing are adjectives.
Also, I think everyone would agree about some specific cases of this; for example, that the word enlightening is an adjective in a phrase like “a very enlightening experiment”.
“an enlightening experiment” is different in some ways from “a running experiment”
You’ve already noticed that we can use a comparative construction with more or most before enlightening (“a more enlightening experiment”), but not before running (*“a more running experiment”). BillJ explains the significance of this better than I do, and Rathony explained it before me. The observation is simply that enlightening is gradable. This is more or less the same thing we test for with the very test (“a very enlightening experiment” vs. *“a very running experiment”).
As Rathony says, the gradability test with very is unfortunately inconclusive here. Only adjective phrases are gradable in this way (I think everyone agrees on this point), but not all adjective phrases are (easily) gradable. So the gradability test can be used to prove that something is an adjective, but it can’t be used to prove that something is not an adjective: a negative result doesn’t tell us anything for sure. Running in “running experiments” does not seem to be gradable, but it still might be an adjective or a verb.
There are also some differences in how we can use enlightening and running after verbs. BillJ points out two parts to this:
[enlightening] can occur as complement to both complex intransitive verbs and
complex-transitive verbs: (It became quite enlightening; I found it
quite enlightening). Other examples are a very entertaining show; a
very frightening film.
I think another complex intransitive verb phrase like “become” may be “stop being”, as in “The experiment stopped being enlightening” vs. *“The experiment stopped being running” (Google Ngrams doesn’t return any results for “stop being running”).
Apparently, verb tests like these show that enlightening is an adjective. But unfortunately, as with the gradability test(s), it’s less clear if failing the verb tests means that running is not an adjective.
Apparently, there is such a thing as adjectives that can’t be used predicatively, including some participial adjectives such as alleged.
It doesn’t quite work to say that running is always a non-predicative adjective, since it can be used in sentences like “The experiments are still running”, but it seems like it could be an adjective of another type, neither always able to be used predicatively or never able to be used predicatively, but only able to be used predicatively in certain circumstances.
In fact, I found a fairly recent paper that says that non-gradable adjectives generally cannot be used as the complement of “seem”. “The Category of Participles”, by Björn Lundquist (2013), cites an argument to this effect by O. Matushansky in “Tipping the scales: the syntax of scalarity in the complement of seem” (2002). Lundquist gives the following example
(12) a. This music seems nice/*choral.
b. This problem seems
insoluble/*mathematical.
(from Matushansky 2002)
It’s a bit difficult to make this clear however because of something called “scalarity coercion”: basically, most adjectives can be gradable, given the right circumstances, even if they are normally non-gradable. The example of this that Lundquist gives is
(13) a. This music seems almost choral.
b. This problem seems pretty much mathematical.
(from Matushansky 2002)
(Lundquist’s main argument is that all participles are actually adjectives, but I’ll get into that later.) Anyway, if Matushansky’s argument is correct (I’m not sure that it is, because the issue of “scalarity coercion” makes it difficult for me to judge the grammaticality of these examples), it seems like the same explanation might apply for complements of verb phrases like “become”, “find”, “stop being”. That would mean that the verb tests don’t give us much, if any more information relative to the very test or the comparative test: it might just all be about gradability.
Interpretations
So far, I’ve just been summarizing the data and facts that I am aware of that seem relevant (in a probably long-winded manner). How to interpret these facts seems to be controversial. I’ve read two papers so far and encountered at least three different approaches.
Bresnan, as presented in Laczkó: “running” has to be an adjective when it comes before a noun, but it could be a verb in some other context
The first paper I read was “Another look at participles and adjectives in the English DP”, by Tibor Laczkó (2001). He cites Joan Bresnan (unpublished) as saying that smiling in phrases like “a smiling child” is an adjective.
This follows from a diagnostic principle that Laczkó presents as follows:
Adjectives but not verbs (including participles) can be prenominal
modifiers: A N vs. *V N.
This diagnostic isn’t obvious to me personally, but it seems to have some appeal to many people. Laczkó in fact says in a footnote that
It is important to note that this view is quite dominant in the
literature, whether generative or non-generative. For instance, Levin
and Rappaport (1986) appear to share Bresnan’s empirical
generalizations, but in their GB framework they offer an analysis
different from Bresnan’s LFG approach. (Here I cannot discuss and
compare these two accounts.) In addition, Ackerman and Goldberg (1996)
also consistently talk about English “deverbal adjectives based on
past participles and used attributively”. (footnote 7)
As far as I can tell, tchrist’s answer classifies running as an adjective due to something like this principle.
Incidentally, Laczkó also mentions another useful diagnostic test from Bresnan: only adjectives can be preceded by the negating prefix un- (but don’t confuse this with the rarer and less productive reversative prefix un-, which attaches to verbs such as undo, unlock). But I don’t think we can say that all adjectives can be preceded by un-, so it doesn’t seem like a negative result is conclusive. Checking unenlightening and *unrunning on Ngrams shows that this is another test for adjective-hood that gives a positive result for enlightening, and an inconclusive result for running. (I don’t know if this tests for the same thing as the gradability tests; I’m inclined to think that it does not, since we can very easily say things like unmarried.)
Under an analysis like Bresnan’s (as represented in Laczkó), a term like “participial adjective” would be appropriate.
Laczkó: verbs can occur pre-nominally or post-nominally, just like adjectives. “Running” could be a verb here
Laczkó’s own view seems to be presented in the following passage:
I would like to make the following assumptions and generalizations.
A)
Prenominal modifiers are not necessarily adjectives: they can also be
participles; and certain postmodifying participle-looking words must
be taken to be participles in any analysis.
B) Thus, it is not
possible to capture the use of all prenominal participle-looking words
in NPs by the help of a single general rule [...] We need two (sets of)
rules: one for the use of -ing and -en participles and another for
participle → adjective conversion.
C) The relevant generalizations for
-ing forms are as follows.
Ca) Participles derived from any one of the three major verb types (unaccusative, unergative and transitive) can
be used in NPs, cf.:
(34)
a. I couldn’t catch the tile falling from
the roof.
b. I couldn’t catch the falling tile.
c. The boy shouting at
that girl used to be my friend.
d. The shouting boy used to be my
friend.
e. The boy reading a newspaper used to be my friend.
(3.2. Participle-looking words in English NPs revisited)
Also:
In this paper I have challenged the quite general view that the
overwhelming majority of participle-looking words in English NPs
(including all the prenominal ones) are adjectives and claimed that
they are participles (non-finite verb forms). (4. Concluding remarks)
His arguments include the very test, the acceptability of phrases like “a crying child” alongside the unacceptability of sentences like “*The child was too crying last night,” and the fact that English attributive adjectives can occur post-nominally as well as pre-nominally (suggesting that there is no absolute link between adjectival status and pre-nominal position). I didn’t find any absolutely compelling argument that e.g. would convince someone devoted to another theory, but Laczkó’s viewpoint makes more sense to me than Bresnan’s (at least, as he presented them; I haven’t read any of Bresnan’s actual writing). As far as I can tell, it also corresponds better to the positions outlined in BillJ’s answer, and Greg Lee’s answer based on Chomsky, which makes me a bit confused about Laczkó’s statement that the “dominant” view in all linguistic literature is that pre-nominal -ing words cannot be verbs.
Anyway, from Laczkó’s perspective, I believe “running” in “a running experiment” would be classified as a participle and as a verb.
Lundquist: all participles are always adjectives and never verbs
I alluded to this earlier, but it’s a bit difficult for me to explain Lundquist’s position because unlike Laczkó’s, it does not mesh with my first thoughts about this topic. But, I found Lundquist’s paper thought-provoking. As I mentioned earlier, it seems to be much easier to prove that an -ing word is an adjective than to prove that it isn’t. Lundquist’s view is that all participles are actually adjectives; it’s just that some are gradable and some are not. The way he explains this is by saying that some participles contain an “event structure” (these generally correspond to what others identify as verbal participles) and others do not (these generally correspond to what have traditionally been identified as adjectival participles or participial adjectives). Somehow, the presence of an event structure is incompatible with gradability. It’s a bit much to wrap my head around (Lundquist apparently works in a “non-lexicalist framework”) but there are some useful factual observations that are actually relevant no matter what theoretical position you take. Lundquist notes that
the restriction on attributive participles is much less strict than
the distribution of participles following e.g. seem, as shown in (7):
(7) a. the recently made headway—all that headway was/??seems made in
a day.
b. the most recently taken photos—these photos were/??seem
taken recently.
c. the kicked out guests—they were/??seem/??seemed
kicked out.
As shown in (7a), even idiom chunks can appear in prenominal
participle phrases, which we can take as evidence that the participle
has a phrasal source, and can thus not have been derived in the
lexicon (see Kratzer 2000 for discussion). It is not obvious that
there is any restriction at all on so-called “verbal” participles to be
used as prenominal attributes. [...] there is no reason to assume
that only “adjectival” participles, i.e. participles that are
event/argument structurally reduced, can be used as pre-nominal
attributes, at least not in languages like English and German (though
it could of course be the case that some languages only allow
adjectives with certain scalar properties in pre-nominal position).
(pp. 3-4)
From Lundquist’s perspective, it seems “running” would be classified as a participle and as an adjective (but it seems it would be an “event structure adjective” in particular, and Lundquist writes that “there are no lexical event-structure adjectives” (p. 11) so he would probably say it is not derived in the lexicon).
References (also linked to in-text)
Laczkó cites unpublished (at the time) work by Joan Bresnan, and Lundquist cites “Tipping the scales: the syntax of scalarity in the complement of seem” (2002), by O. Matushansky.
Further reading I haven’t gone through yet
These sources were not used in the writing of this answer; I am linking to them because I encountered them later on and they seem relevant.
“Adjectival present participles: an aspectual constraint”, by Aya Meltzer (I’ve looked at it a bit and would highly recommend reading it; it describes many, maybe all of the tests I mention here, and also a number I didn’t know about)
“Participles, gerunds and syntactic categories”, by John Lowe (2016)
“Remarks on gerunds”, by James P. Blevins (2003)
“Word formation is syntactic: adjectival passives in English”, by Benjamin Bruening (2014). This discusses issues of syntax vs. lexicon, and also has an appendix that talks about the word-class categorization of participles (in this case, passive participles, but I think a number of the arguments would be applicable to active participles also)
ongoing, a close synonym and a more evident adjective. You still can't get rid of the absurdity.