Wow I never expected this topic to be so big. Guess there were more anime watchers overlapping with the gamers here than I thought. Cool to see everyone contributing!
Once again some other favorites that people have mentioned:
Big O, I really enjoyed the art style and plot on this one
Evangelion, of course it's a classic. One of the deepest animes ever made. If you want something to really make you think, this is your show.
City Hunter, loved the plot.
Princess Mononoke, Howl's Moving Castle, Spirited away: Hayao Miyazaki is brilliant. It's as simple as that. If you haven't seen these movies, do so immediately.
Bubblegum Crisis, The classic cyberpunk anime. I'll always have good memories of this one.
Detective Conan, I've always been a sucker for a good mystery, and this show still keeps pumping them out. I can't believe how long it's gone on.