"THE EMPEROR PROTECTS!", or anytime Shinji's imaginary friends save him from his self-doubt. Some of the Maya-Ritsuko moments (not those kind) are "awwww"-worthy, too. There's also the scene between Rei and Shinji in the Alternate Universe omake, which is so cute that it gives Asuka a nosebleed.
Shinji earning Rei's eternal devotion at the end of the battle with Ramiel. Also insanelyawesome, and gives Ritsuko a headache trying to figure out how they managed it.
