- Legend of the Galactic Heroes: Yeah, with a name like that, you'd think it was some bad 80s shounen fighting show, but no, it's actually an incredible - albeit really long - space opera with lots of characters, politics, strategy, and philosophy. Unfortunately, most studios consider such a long and long winded anime to be unmarketable to American anime fans, so you're pretty much stuck watching it fansubbed on youtube.
- Ghost in the Shell: Let's face it, very few directors can get away with having the main character naked a good third of the time and still be really smart, but Mamoru Oshii is one of them. Then again, most anime fans have already seen it.
- Akira: A drama, wrapped in an allegory, wrapped in a social commentary, wrapped in plastic explosives. Again though, it's kind of anime canon.
That's all that comes to mind right now, but this seems like a good place to start. I'm actually tempted to add Neon Genesis Evangelion to the list, but the thing is, even though the characters are some of the smartest written in any television show or movie I've ever seen, and even though it does actually have a lot to say about the human condition, it's not deep for any of the reasons people seem to think it is. All of the genuinely interesting things it has to say are wrapped up in a metric shit-ton of bullshit pomo existentialism, bullshit pop psychology, and easily misleading imagery. While all that does give NGE a cool atmosphere, it has the same problem as 2001 A Space Odyssey, where it doesn't so much provoke thought as it makes the audience go "ooh, this is so deep." Still, for the character drama alone I'd recommend it, and like I said, beneath all the BS, there actually is some pretty deep stuff, so give it a shot. You might love it, or you might hate it, but after all, how can I fail to recommend my favorite anime?