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.