What is the structure in "as best you can"?
Solution 1:
"As best you can" is an idiom with archaic syntax. A more modern version would be "as you can best", though that is still not particularly normal today.
"As well as you can" is a different construction.
As various people have said "as best as you can" is a blend of the two constructions (with the opacity of "as best you can" contributing to the confusion).
[Made an answer on drmj65's request.]
Solution 2:
Best is the superlative not only of the adjective good, but also of the adverb well and has the meaning, in the OED’s definition, in the most excellent way, in the most eminent degree; in the most suitable manner, with the greatest advantage, to the fullest extent. As best you can means in the best possible way you are able. You might well wonder why it isn’t as best as you can. After all, we say as well as you can and, as David says, as tightly as you can. I can’t give you the answer to that without further research, but I cannot agree with David that it is in any way incorrect or 'sloppy'. It is an established part of the language and well attested across several centuries. The OED has numerous citations illustrating it. Here just three examples:
She . . . bundled up her hair as best she might (1862).
When they died she callously got rid of their bodies as best she could (1870).
Unable to stop himself, he simply moved off and away, leaving her to contend with Al's homage-laden stupor as best she might (1998).
Solution 3:
This is a common expression (with the second "as"), but the logically correct version is "as well as you can."
Without the second "as," this is elliptical, meaning it has words left out: "as." In other words, it represents sloppy language use.
Would you say: Pull the rope as tightly you can? No. However, this odd example sentence has the same structure as that which you are proposing. You would write "as tightly as you can."
Even so, "as best as you can" doesn't make logical sense. You could say "give it your best effort" or "to your best ability." The reason why it doesn't make sense is that "best" is a superlative term. Something is the best or not; "best" isn't subject to qualification, as this phrase attempts.
Solution 4:
This dictionary defines adverb as
a word which describes or gives more information about a verb, adjective, adverb or phrase
Best is the adverb in "as best you can." It illuminates "as you can" (as you are able) to be "the best you are able."
Solution 5:
I suspect, though I don't have my usage books with me, that this is a case of one idiom influencing another. The constructions that "sound right" to a native ear in American English are either "do the best [that] you can" or "do as well as you can". In the first case, "the best" is a noun, the subject of the verb "do" (in the imperative) ("what will I do?" "I will do the best [that] I can"). In the second case, the "as well as" phrase seems to be performing as an adverb modifying "do" ("how will I do it?" "as well as I can.") It's certainly true that "do as best you can" is well attested, but I don't think most Americans would consider it standard usage. As far as grammar goes, superlative or not, it is "missing" the second "as", standard in such adverbial constructions. I imagine that in use the "as" was "swallowed up" by the "best" ("best as" becoming "best's" becoming "best").