Asuka, and later Shinji, learning how to drive. Even better, the pick-up they were using was a standard stick-shift. It's a wonder that they didn't hit anything or burn out the transmission.
Shinji's attempts at catching chickens.
A policeman and Rei are trying to find out who Aki's parents are, but Aki doesn't know their names (since she always just calls them "Mama" and "Papa"). The policeman asks what her mother calls her father other than "Papa":
Aki: *thinks hard* "Baka!"
Policeman: "Well, that narrows it down to just about every married man in the city..."
When Asuka tries to get Aki to say "Papa", we get this little gem:
Mama: "Can you say 'Papa'? 'Pa – pa."
Papa: "...She didn't really just say what I think she said, did she...?"
The mental image of Shinji following a young Aki around everywhere with a video camera is just adorably hilarious. And he still misses the part (her first steps) that he was really hoping to see.
When Misato tries to tease Asuka by suggesting her a man's thong as a present for Shinji, Asuka tells her about a time when he actually attempted to surprise her by wearing one... which promptly sent her into a fit of helpless laughter.