RAHXEPHON!
Sorry, I get excited. Seriously, though, RahXephon is probably one of the best anime series of the last decade. Even aside from the absolutely superb voice acting and animation, the characters are compelling, and the story is fantastically written. It's not often that I feel as emotionally attached to the characters (chalk it up to Monica Rial, Chris Patton, Hillary Haag, et. al being fantastic, in addition to everything else), especially the "bad guys". Not many series exist which give their antagonists a real backstory, rather than "Draco-in-Leather-Pants badass". It has some mind games (and watching it a second time, knowing everything you learn toward the end is a real treat), but it's never out of control, and feels like the hints are really supposed to be understood.
Utena has some good psychological games, and excellent replay value. Watch it as an allegory for feminism, or like it's about overcoming psychological trauma, or different parts as representative of different things (like retrieving a sword being a symbol of sex). Again, the psychological elements are subtle, but clear. It does some mind-fuckery, but never to the point that it gets so wrapped up in it that it loses perspective and storytelling. Watch it, for it is good.
Evangelion. More a right of passage than anything else. You have to watch it if you want to claim any kind of props for watching anime, and it's a common experience most of us have. If you remember the entire time that the writer/director pretty much went bat-shit-insane over the course of the series, and that there's no intentional deeper meaning, it's much easier to swallow. If you go in for the mind-bending, prepare for something which will either leave you nodding and contemplative, or screaming "what the hell just happened" every fifteen minutes.
Nadesico. Taking the piss out of Evangelion (and most other giant robot anime) one step at a time. It's all of the tropes, and could easily have been done as pure parody (a la Excel Saga. But it isn't, and that makes it even better. Some moments of pure comedic brilliance, balanced against an honestly intriguing plot, and characters who compel you to care about them. Did I mention that it'll leave you laughing for minutes, and having to rewind? Watch it, preferably, in English, since most of the jokes were translated to be better in English.
Full Metal Panic. PANIC. Sorry, again, for the emphasis. I hate people who think I misunderstood the name of Full Metal Alchemist. Two different shows, I promise. FMP is among the best giant robot shows I've ever seen, mostly because the giant robots aren't the focus of the show (see also: Nadesico, RahXephon). When the robots are used mostly for "hey, it's cool" and "incidental to the rest of the story", you get the best of both worlds; you get all the big-gun action without losing the human element of the story. Sousuke Sagara is a fantastic character, and watching his growth through the series is a real treat. As before, you really do feel an emotional connection to the characters, and some of it is damned funny. Lots of fish-out-of-water humor around Sousuke being a military man thrust into civilian situations.
D.N Angel. Okay, it's pure sitcom. It's pure romantic comedy. But it's a lot of fun. Yes, it's set up with a bunch of teens. Yes, it's a lot of angst. But, the characters are worthwhile, and it's a very sweet story. Nothing really big by way of thought-provoking, but it's nice. Voice-acting was well-done, the localization worked brilliantly.