The New Testament probably

Unfortunately for everyone alive, it's probably Harry Potter and the Methods of Rationality

So "fan fiction" is a bit of a hard thing to label any myth pre-20th century and intellectual property/copyright nonsense, but if we take "fan fiction" to be very loosely "a story that uses characters from another story" then the answer 100% has to be The Odyssey, if we're limiting it to things that we know exist and have written records of. Scholarly consensus is that The Iliad and The Odyssey, which in classical Greece were both considered to be by an author named "Homer," were actually created by the different poets, and that The Iliad is the older of the two. So The Odyssey, which is the return home of one of the characters of The Iliad and indeed the basis of like every "adventure" story made in the Western world, has gotta be the most influential extant fan fiction. That said, really all Indo-European cattle raid/warrior type stories (think The Iliad from Greece, Mahabharata from India, Táin Bó Cúailnge from Ireland) are all fan fictions of an original proto-Indo-European story of that nature, but we have no written records of it existing, we can just sort of assume that it does based on similarities between all of these stories.

My Immortal

Paradise Lost.

Trans Wizard Harriet Porber And The Bad Boy Parasaurolophus

50 shades if we're being strict about the definition of fanfiction. If we go full chaos everything is fanfiction mode, maybe the bible?

I dont know about of All Time, but the most influential fanfiction of recent memory is the Chris Chan SonicChu Fanfic. The author is supposedly "the most documented person in history" and the culture around it led to the creation of KiwiFarms and a whole lot of other awful nazi shit

Fallout: Equestria is some pretty amazing fanfiction if you've never heard about it

Diane Marchant’s 1974 Kirk/Spock story “A Fragment Out of Time.”

The first slashfic

Sexy times with Wangxian

The Phantom Menace

The Lusty Argonian Maid