I can highly recommend
Neon Genesis Evangelion: An absolute classic of anime, it's a giant robot sci-fi thing, but if that's all you get from watching it then you've missed the point. Evangelion is a deeply philisophical show, and focuses particularly on human interrelation, with frequent allusions to Sigmund Freud. I've seen it four times, and enjoyed it all four times, it's insigntful, and it's just an all around great watch.
Mushishi: This was just released on region 1 DVD, so it's finally legally available in the west. Mushishi honestly is kind of hard to describe, because it's so different from any other anime I've ever seen. It's a quiet contemplative series of stand alone stories about a traveling doctor who treats illnesses created by supernatural creatures called mushi. If you require constant smashey smashey, it's not the show for you, but if you're willing to give its sober approach to entertainment a try, I doubt you'll regret it.