"My hardest" in "try my hardest" is probably not a direct object
I think "my hardest" is not generally a direct object when it comes at the end of a verb phrase like this, because in all cases it fails the passivization and extraction tests, and in some cases it can co-occur with what appears to be a direct object.
I don't know if this is conclusive evidence for the precise grammatical role of "my hardest" in expressions like this, but it seems sufficient for me to think that it is not a direct object.
Examples of "my hardest" in a verb phrase containing another phrase that seems to have the role of direct object
A Google search for "hit it my hardest" turns up a number of examples:
I wasn't trying to hit it my hardest, I just wanted to get it over
Taryn Hicks, "0/12/13 - Troy Tournament"
I learned that I didn't have to hit it my hardest every time,” Chisolm said.
"2016 Volleyball Co-Player of the Year: Claudia Chisolm, Calvary Day"
To me, it seems pretty apparent that "it" is the direct object of "hit" in these examples; unfortunately, I wasn't able to find passive versions of "hit it my hardest" (like "It was hit my hardest") so I guess it's still possible to try to analyze it elsewise, if you can figure out how to do that.
I did found three examples of the passive "I was pushed my hardest" on Google, which doesn't quite seem to be the conventional usage of "my hardest" but which is relevant I think.
It's possible that "hit it my hardest" is a different construction from "try my hardest," but I think it's more parsimonious to assume that "my hardest" plays the same grammatical role in both.
In fact, even though they seem non-standard to me, I did find a couple of examples of "try it my hardest," which I think serve as evidence that at least some speakers do not think of "my hardest" as the direct object of "try" in this construction:
And I to promise to try it my hardest." Tears welled up in Rini's eyes as she wrapped her arms around my and Darien's neck.
"Reversed Event", by Twi-chick34 | fanfiction.net > Anime/Manga > Sailor Moon
Promise I tried it my hardest
Taj Ahkel – On Two Feet Lyrics | Genius Lyrics
Two failed direct object tests
The passivization test fails, of course:
- *My hardest will be tried.
I cannot figure out a way to extract "my hardest", so I would say it also fails an extraction text.
- *My hardest is what I will try.
The non-conclusive evidence I know of for it being a direct object
- Position: it's in the right position for an object
- Noun phrase grammar: as StoneyB mentioned, this kind of superlative indisputably has the structure of a noun phrase internally, which is why it can take the determiner "my."
- Rarely occurs alongside a direct object: When I looked through Google Books, all of the instances I found of "my hardest" were after verbs that did not have another noun phrase that could be interpreted as the direct object. For example, run my hardest,
try my hardest [to _____], work my hardest, paddle my hardest, fight my hardest, pitch my hardest, strive my hardest [to _____], pull my hardest, pray my hardest. There were no Google Books results for "hit it my hardest" or "was pushed my hardest". So these constructions with an object in addition to "my hardest" are evidently quite rare (although, as mentioned above, there are examples from Google web search).
Failing the passivization and extraction tests probably is not enough on its own to prove that "my hardest" is not a direct object in sentences like this. To me, the so-called "cognate object constructions" seem similar, and if the name is accurate they are considered objects in at least some analyses. These are constructions like "They died a gruesome death," "He smiled an evil smile" or "I slept a good night's sleep."
But many "cognate objects" fail the passivization test or an extraction test: we definitely can't say
- *A gruesome death was died by them.
and I also don't think
- *A gruesome death is what they died.
If "my hardest" is not a direct object, what is it?
I don't know what other options there are for analysis besides "direct object." As other answers have pointed out, "adverbial" seems plausible.
In fact, the OED quotation cited in ermanen's answer seems to follow this analysis for the similar expression "one's best" in at least some cases, since it says
In later use one's best is sometimes used adverbially with verbs of performance other than do and try and has the sense ‘as well as possible, as best one can’
While this does make an exception for do and try, as I said earlier, I don't see any clear motivation for doing that.
I've also run across the term "pseudo-object" in "Leg it, floor it, snuff it: A synchronic and diachronic analysis of dummy it" (Mondorf; see also Mondorf 2016 which may also be relevant) but that doesn't seem very explanatory, just a bare description.