I'm not a die hard fan but I do rather enjoy anime. I was introduced to it courtesy of my INTJ friend who has a huge collection of them and loves having movie nights.
Of the few I've seen, my favourites are (in no particular order):
My Neighbour Totoro
Evangelion 1.0 and 2.0
The Girl Who Leapt Through Time
Death Note (although I'm only 1/2 way through - don't ruin it! )
I need to see Perfect Blue and Akira again because they were the first animes I saw and they freaked the hell out of me. I'm thinking that maybe I'd feel differently now...