Post-Third Impact fic. This can range from Shinji and Asuka by themselves in a wasteland, to everyone coming Back from the Dead and the world returned to (mostly) normal. Most of the time, Rei has vanished and Toji gets his leg back (both of which might be supported by canon).
There have been several stories involving Shinji being raised by someone other than his uncle/teacher and exploring the For Want of a Nail aspect of how Shinji's personality changes. Sometimes Shinji gets raised on a product, like the aforementioned Shinji And Warhammer 40K.
Reincarnation fics, especially Kaworu/Shinji Reincarnation Romance. As well as explaining how the different canon continuities are related to each other, this gets rid of the pesky "Kaworu is an Angel" problem and lets poor Shinji be happy for once. They are especially popular since the third movie of Rebuild of Evangelion.