Your phrasing implies that you're thinking specifically of Greco-Roman texts, so first, a minor correction: we owe virtually everything we have of them to Christian monastic and scholarly traditions (Latin) or to Byzantine scholarship (Greek). Only for a tiny proportion of texts -- and they tend to be rather obscure ones -- do we rely on Syriac, Ethiopic, or Arabic (Islamic) sources.

The vast majority of "original" documents that we have are archaeological finds of papyri, usually very fragmentary ones. A very large proportion of these texts are scraps of just a few letters or words. (After the year 200 CE or thereabouts we start to see parchment as well, but it's a very similar situation.) The rest is accounted for by inscriptions, usually in stone, occasionally in metal.

All of these would fit your question as posed. A fairly large number are written by the hand of the authors themselves, seeing as the papyri are mostly domestic or legal documents of one kind or another (letters, contracts, etc.). But only a very small proportion of them are copies of texts that you'd see in "Introduction to Greco-Roman Literature 101". Still, over thousands upon thousands of papyri, that still comes to a fairly large number of papyri; the Homeric Iliad is far and away the most represented, with at least 1569 copies known (as of 2001), mostly from the 2nd to 4th centuries CE.

We don't have any copies of texts by a well-known author written by the hand of that author until Flavius Dioscuro in the 6th century CE, according to this post. /u/georgiusflorentius also cites an earlier papyrus from the 1st century (covered by this 1991 article, if you have access) but though that papyrus -- and the others listed in the 1991 article -- are apparently autographs, they're not by known authors.