Not in their own hand, of course ... but when it comes to people like Horace, Virgil, Tacitus, Cicero, etc., do we have any copies of their works that were published either in their own lifetime, or shortly after?
Because I've always wondered if we're just relying upon copies of copies of copies of copies, and so on. Were the humanist scholars of the 14th Century just copying Carolingian documents? Or did they have original material from Roman and Greek times?
For the most part, yes, surviving copies are probably at least tenth-hand. Typically they will have been copied several times through the Mediaeval and Renaissance periods, and presumably copied a similar number of times in antiquity. Most Greco-Roman texts are known from 14th-to-16th-century manuscripts; if we're lucky, we have some earlier ones too. The most important task once you've got your manuscripts is to work out the chain of dependence between them.
A trivial counter-example would be non-literary papyri (letters, accounts, contracts, etc.): these are of course autographs. But I guess that's not the kind of thing you're thinking of.
I'm not aware of any literary papyri that date to within the lifetime of the text's author -- that doesn't mean there aren't any, of course. The nearest misses among published texts that I can think of are the following. In each case there's a few decades' uncertainty in the date of the papyrus and/or the original text.
When you get into Byzantine texts, we've got a fair number of manuscripts written in the author's own hand or with annotations in the hand of a well-known author -- people like Photios and Arethas (9th century).
Inscriptions are another area where you can guarantee a text's contemporaneity with a well-known historical figure. This includes Bronze Age records in Egyptian and Hittite archives. These can include semi-literary inscriptions, since many Greek inscriptions are in verse: the earliest example I know of is Inscriptiones graecae I^3 784, a hexameter (epic) inscription which commemorates Athens' victory in the Battle of Marathon in 490 BCE, and which records its dedication by Kallimachos of Aphidnai, one of the two Athenian generals in the battle.