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The best reincarnation fanfic I've read here is probably Soaring on Little Wings by tom117z.
It's a very well done fic in which the mc, who was a small kid, dies and finds herslef in the body of a little pegasus filly, who is adopted by Rainbow Dash's parents on Cloudsdale.
It's very well written, with excellent pacing and story development. As you follow Little Wing's life, from filly to a young teenager to adulthood you cannot help but feel connected to her.
Really recommend it.

For the NON-fanfiction, the choice is obvious: To Your Scattered Bodies Go by Phillip Jose Farmer.

Hm.... Reincarnation fanfic that is *not* "I died and showed up in Equestria" or such. Does it count if the character is reborn as *themself* perhaps? I've done two of those, from The Dance of the Sun to In Tambelon Comes The Hungry Darkness. Hm... I really haven't read enough of those to have one that sticks in my head. Technically, Project: Sunflower would count, since Erin 'dies' and is reborn as Sunflower, and it is epic.

The field of "Alicorns are immortal because they respawn" fiction needs more stories.

The familiar feeling of aetherial decoupling and recoupling swept across Celestia before any pain, so the tragic accident was really nothing compared to some of her previous deaths. Still, when her material body reformed in the study and she saw a panicked Twilight Sparkle bending over the corpse of her beloved princess, it brought another pain to her immortal breast. She really should have told Twilight, or at least been more cautious with the spells she had been taught.

"Twilight," began Celestia with the greatest of caution, "please do not--"

With an anguished shriek, Twilight spun in place and her horn flared incandescent. This time, Celestia felt part of the fire that took her life, and after the familiar sensation of rebirth passed, it left her slightly peeved.

"Twilight!" she snapped. "Stop this--"

By the time Celestia could calm down her panicked student, there were a full dozen alicorn corpses in the study, displaying various levels of incinderation and exsanguination, but all of them part of the disposal problem that Celestia was tempted to just leave to the castle staff and to heck with the consequences.

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I suppose it all comes down to what you count as technical reincarnation. As Georg pointed out, Project: Sunflower could count, despite being "reborn" into an adult body that is in the shape of a pony (though not actually a pony until she goes through the process again later on). There are quite a few stories where people die or nearly die, and get put into the body of an adult pony. Sometimes, it's not so much as being reincarnated as just having Equestria as their afterlife.

If you stretch yours expectations of reincarnation (and not just being reduced to having to go through infancy again with memories of their last life), a ton of pony transformation stories count along those lines. Some of them are even quite good with similar themes to reincarnation stories.

Stories where it's a pony that dies and is reincarnated is much, much rarer. I can only think of a few off the top of my head, and half of them are played straight, the other half are alicorn reincarnation stuff, which is basically they get a new body. If they are lucky, they don't have to go through foalhood again, though some of them do.

Also, almost all changeling transformation stories are played like reincarnation stories, since they are podded or even put into eggs as they change into their new forms.

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There's the Tales of the Tutelary Spirits, though the author hasn't updated in a while.