I don't watch a lot of anime, and in fact was previously adamant against watching any, until I was dragged into the world of FLCL, where I fell in love.
So far, I have seen:
Neon Genesis Evangelion
I also just saw the Evangelion 1.11 movie last week, with the second (and End of Evangelion) soon to follow. And right now I am borrowing, from a friend, Paranoia Agent (which I've seen the first episode of) and the Eureka Seven movie.
"Vishnu takes on his multi-armed form, and says, "RAAAAAHHHOOO!!!'"