For me, it’s The Epic of Gilgamesh. I don’t think I’ll be breaking that record anytime soon.
Don Quixote
The Bible. 
Didn’t get past Numbers, though.
The Torah.
The book of Job, though we can’t know when it was written. It’s probably the oldest book in the Bible, putting quite a way back.
Almost non-religious fiction and also the most boring book I’ve ever finished because I had to for school:
The Pilgrim’s Progress From this World to That Which is to Come: Delivered Under the Similitude of a Dream Wherein is Discovered, The Manner of his Letting Out, His Dangerous Journey; And Safe Arrival at the Desired Country.
Modern publishers try to hide this books boring, boring innards by titling it Pilgrim’s Progress.
Beowulf is pretty old too. It’s older than Pilgrim’s Progress but it sure doesn’t feel older than Pilgrim’s Progress.
The Odyssey, The Old Testament, and I feel like one other but I can’t think what it is.
I’ve read The Iliad, the Old Testament, and The Art of War. I think the original dates of all are disputed.
If you’re talking about when it was printed, I’ve read part of the book where Darcy comes up with his law concerning head loss through soils. If you’re talking about when it was written, Plato’s Republic.
I’ve read The Odyssey, The Iliad, and The Aeneid, and The Art of War.
For “Non-classical” literature; probably Sir Thomas Malory’sLe Morte D’Arthur, with the original spelling (The grammar and paragraphing had been cleared up and the spelling had been given some consistency, though), along with Marco Polo’s Travels and Niccolo Machiavelli’s The Prince.
Parts of The Canterbury Tales and The Decameron, in high school. I remember liking both, especially The Decameron.
Plato’s Republic, if that counts.
Gilgamesh. But it’s the oldest literature period. After that, Illiad and Odyssey.
Counting non-literary texts: I’ve read some Linear B texts, though I forget why.
Literary texts: Depending on who has priority, either Homer or Hesiod.
It’d be interesting to consider the oldest book not read in translation. I’ve read bits of the Greek Bible, and a large part of Parzival, but for a whole work longer than a few pages, the oldest for me is Gawain and the Green Knight.
The Greek stuff
(again…not counting the Bible)
Ditto.
The Bible. I’ve read chunks of it not in translation, too - I used to go to a Torah study thing in college that assumed a relatively decent knowledge of Hebrew (which, at the time, I actually had).
Not counting the Bible…The Iliad, I guess.
But everybody’s read The Iliad and The Odyssey, no? I had to read those in High School. Isn’t it, like, required?