This question was inspired by the discussions in this post.
Consider the sentence
 He arranged for me to go there.
What is required is to determine its constituent structure (in the sense of phrase structure grammars). Also required is a demonstration that the proposed structure is correct.
This much I will take to be uncontroversial: He [arranged for me to go there]. In other words, the words in the brackets are a constituent, namely a VP.
But beyond that, there was a bit of a controversy. Some have argued (in effect) that arranged for is a constituent. I and others have argued that for me to go there is a constituent.
Note that these two claims are inconsistent; we can't have overlapping constituents. What can happen is that the structure of the clause is ambiguous, though usually that is accompanied by a difference in meaning. I do not think that arranged for is a constituent of  (at least not in the most obvious interpretation of the meaning of ), but if you think you have an argument that it is, this will be of interest.
My original argument that for me to go there is a constituent
I originally gave two arguments that for me to go there is a constituent.
Here is the first one (note that it will be called into question below!) Consider
 He wished for me to go there and arranged it, too.
The second argument uses coordination. Consider
 He arranged a nice dinner and for me to go there.
Usually only constituents can be coordinated. Yes, there are some known exceptions that involve non-basic coordination, but if you rule those out, an argument from coordination can be successful. CGEL's Sec. 15.4.6 ('Coordination as evidence for constituent structure') is precisely about that, and they dedicate two whole pages to working out an example in detail (pp. 1348-1350). I will not reproduce the whole thing here, but one needs to show that  cannot be analyzed as any of the following: 1. a clause-coordination with ellipsis of the subject and the verb, 2. a right nonce-constituent coordination, 3. a delayed right constituent coordination (right-node raising, RNR). I believe I can do all that, but won't write it out right now.
Problem 1: could me to go there be a constituent?
There is a possible problem with the above analysis: a test analogous to that in  seems to suggest that me to go there is also a constituent:
 He wished for me to go there and arranged for it, too.
Now it seems that the antecedent of it is me to go there, and so, by the proform substitution test, me to go there is a constituent. The argument is exactly parallel to the one I gave for the constituenthood of for me to go there.
The problem with that is that to me (and some others, including Greg Lee) this seems very implausible and counterintuitive; it just doesn't look like a constituent. That's all—but it's enough.
And if me to go there is not, in fact, a constituent, then our reasons for thinking that for me to go there is one becomes less certain, because we used parallel arguments to establish their constituenthood.
Let me point out that I could in principle, live with me to go there being a constituent. It wouldn't contradict anything else I've been claiming. In particular, it wouldn't contradict that for me to go there is also a constituent; all it would say is that the constituent structure of for me to go there looks like this: [for [me to go there] ].
Presumably it can be further shown that to go there is a constituent, and then CGEL (pp. 1185) has a general argument that this has the structure [to [go there] ].
In the comments to this answer, Greg Lee has challenged the analysis of  in the following way:
By the same reasoning, is I ... go there a constituent in
 I seemed to go there because he wished for it ?
(Answer: no, because underlying constituents can be broken up by transformations, like Raising.)
Later on, he said that the problem with  is this:
being an antecedent for definite anaphora doesn't show that a string of words is a constituent.
He also said that when anaphora is indefinite, then the substitution test is normally reliable.
Note that if this is a good objection to the test in , it is also a good objection to the test in .
(It turns out there is here is quite a bit of work that questions the assumptions behind the substitution tests, e.g. here and in the references in footnotes 10 and 11 here. However, none of those references seem to be addressing Greg Lee's point.)
So here are some additional tests one can do to establish that for me to go there is a constituent (while hoping that they will not also establish that me to go there is a constituent, too).
Coordination, part 2
But first, what does coordination say about me to go there? Consider tchrist's example:
 ???He arranged for an early dinner and me to go there.
I have to say that I (and some other people) do not find this acceptable at all (though it seems at least some people do). If this is not acceptable, that doesn't in and of itself yet mean that me to go there can't be a constituent, but it's promising that this test doesn't offer any evidence that it is a constituent.
Now let's look at other tests.
Fronting: ?[For me to go there], he arranged. This is a bit Yoda-esque, but definitely much better than *[Me to go there], he arranged for. So this test provides marginal evidence that for me to go there is a constituent, and no evidence that me to go there is.
Heavy-NP-Shift: in our case, this will be a misnomer because what we'll be shifting is not an NP, but rather something that can appear in many functions in which an NP can appear. Consider these sentences: He arranged for me to go there himself; He arranged himself for me to go there. Since we were able to move for me to go there, it is a constituent. In contrast, *He arranged for himself me to go there doesn't work at all. So this is good evidence that for me to go there is a constituent, and no evidence that me to go there is. Note that the move becomes more and more preferable the 'heavier' (i.e. longer) the phrase: He arranged for me to very quickly and without delay go there himself is awkward, while this is much better: He arranged himself for me to very quickly and without delay go there.
Clefting: ?It was [for me to go there] that he arranged. Again, not super-natural, but this doesn't work at all: *It was [me to go there] that he arranged for. Overall verdict is like with fronting.
Pseudo-clefting: ?[For me to go there] is what he arranged. Not great, but better than ?[Me to go there] is what he arranged for, which, in turn, is a bit better than the constructions we've had so far in which me to go there was supposed to work as a constituent. So, weak evidence that for me to go there is a constituent, and marginal evidence that me to go there is.
Answer fragment: Q: What did he arrange? A: For me to go there. I think this is the most natural so far in this section, thus good evidence that for me to go there is a constituent. Now what about Q: ?What did he arrange for? A: Me to go there. Personally, I would probably hesitate before saying that. But perhaps I would say it in the end, and I don't know if I would really pay much attention if someone else said it. So, marginal evidence that me to go there is a constituent.
Passivization [For me to go there] was arranged by him seems fine. The prepositional passive doesn't work: *Me to go there was arranged for by him. So good evidence that for me to go there is a constituent, and no evidence that me to go there is.
Note that this is fine: My going there was arranged for by him. The non-passivized version is fine, too: He arranged for my going there. In this case, for is a preposition taking a non-finite, gerund-participial clause as complement, which prepositions readily do, e.g. They're talking [about my moving to New York]. In contrast, in  and all the other clauses we've been looking at so far, for is not a preposition—at least according to CGEL (see here) and some other sources.)
Parenthetical Insertion: He arranged, hopefully, for me to go there is fine. ?He arranged for, hopefully, me to go there is much less so.
Overall, it seems to me that we have a pretty strong case that for me to go there is a constituent of . The case that me to go there is a constituent is much weaker, but it's not nonexistent.
Is there some better way to show that for me to go there is a constituent (if it is one)? (Better than any of those I presented above.)
Is there an argument that arrange for isn't a constituent? (Is there one that it is?)
Is me to go there a constituent? If not, can a reason be given? (Note that failing a constituency test is in general not evidence that something isn't a constituent. Failing any one such test—or even failing all of them—merely says that the question remains open.)
Can someone clarify the problem with the substitution test (definite vs indefinite anaphora)? I googled for it quite a bit, but couldn't find anything. Is there a way to make the substitution test work to show that for me to go there is a constituent (preferably in such a way that there isn't a parallel argument that would show that me to go there is a constituent)?