Since Hamlet was published in several editions during the Jacobethan era, it is worth looking at how these early editions rendered these lines.
The first quarto (Q1), published in 1603, which has sometimes been called a "bad quarto", gives the lines as follows:
O that this too much grieu'd and sallied flesh
Would melt to nothing, or that the vniuersall
Globe of heauen would turne al to a Chaos!
The second quarto (Q2), published in 1604, gives the lines as follows:
Oh, that this too too sallied flesh would melt,
Thaw, and resolve itself into a dew!
Finally, the First Folio, published in 1623, gives the lines as follows:
Oh that this too too solid Flesh, would melt,
Thaw, and resolue it selfe into a Dew:
ShakesearesWords.com gives two meanings of "too":
- (adverb) very;
- (adverb) anyway, in any case.
There are two other meanings recorded in C. T. Onions's A Shakespeare Glossary:
- adv.: "and too" And at the same time. "wild, and yet, too, gentle* (The Comedy of Errors 3.1.110).
- prep. (var. in F. of) To. "if they come too't" (Hamlet 4.5.60).
Based on this, the most plausible meaning of "too too" is an intensification of "very". So what do modern editions of Hamlet say?
The modern editions I have consulted comment on "sallied" versus "solid" but not always on "too too".
Bernard Lott (The New Swan Shakespeare, Advanced Series, Longman, 1968) says,
too too solid: This is how the First Folio reads. Other early editions, the Quartos, give the word sallied instead of solid, which in Shakespeare's English can mean 'troubled'. But solid contrasts with melt, and dew in the following line, and is probably the correct reading. Hamlet is speaking of the flesh of his own body.
T. J. B. Spencer (The New Penguin Shakespeare, 1980) chose "sullied" with the following rationale:
Q2 reads 'sallied' (which could be a spelling of sullied). F reads 'solid', which contrasts well with melt, | Thaw, and resolve itself... and until the twentieth century was generally preferred by editors. But it may have an unpleasantly comic effect, especially if Richard Burbage, the actor who first played Hamlet, were putting on weight (compare He's fat and scant of breath, V.2.281). Sullied fits well into the feeling of contamination expressed by Hamlet; and for sullies (F 'sulleyes') at II.1.39. Q2 has the spelling 'sallies'.
G. R. Hibbard (The Oxford Shakespeare, 1987) glosses "too too":
altogether too, much too. The reduplication of too for emphasis was very common from about 1540 to 1660 (OED 4).
(Hibbard comments on the two lines in an appendix on pages 382-384, which also discusses John Dover Wilson's theory, already mentioned by Spencer, that "solid flesh" would have provoked mirth due to Burbage's girth [I couldn't resist the pun] but without agreeing with it. His choice for "solid" is based on other arguments.)
Hibbard's gloss appears to settle the issue.
One final comment: the difference between the first quarto and the other editions is the type of deviation that once gave rise to theories that this "bad quarto" might have been a pirated text, either by someone in the theatre writing down the playtext in shorthand during a performance or some actors of minor roles restoring the playtext from memory. Such theories no longer enjoy general support among Shakespeare scholars. The more interesting observation is the wording "too much", where the second quarto and the folio have "too too"; this also suggests that the repetition of "too" was intended as an intensifying device and not, as Alexander Sloane's answer claims, that one of these adverbs here means "also".