*ahem*
Lupin III : Castle of Cagliostro (Miyazaki's first movie, about a group of thieves that are modernly considred anime classics and are inspirations for many works) For 1978, this is some very clean animation and very awesome work all around. Really hard to believe the time it was made.
Ghost in the Shell Stand Alone Complex: Anime based off the films / graphic novel, pretty great.
Neon Genesis Evangelion: Classic action anime that has a lot of psychological, specifically Freudian, undertones toward the end.
Aim for the Top!Gunbuster : Oldschool 1980s anime, made by a company that was just starting out, because of this art quality is a mixed bag but it plays with a unique mix of realistic space travel concepts with action, and also mixes the very emotional elements of Shoujo (girls comics) with the action elements of a Shounen (Boys comics.) There's both a movie version and a 6 episode series.
Aim for the Top 2! Diebuster: Sequel to the above. There's both a movie version and a 6 episode series. When hand drawn it looks amazing, when 3d is incorporated (which is rare but not rare enough) it looks odd and out of place, but if you can ignore it for the story it's great.
Click here to see "making of" animation key frames. I <3 Gainax.
Fullmetal Alchemist Brotherhood: Still airing in japan, you can catch the episodes here:
http://www.youtube.com/watch?v=iMw9OdKv-uY Based on the manga, about two boys presuing the arts of "alchemy" and trying to find a way to restore their original bodies they lost because they broke a forbidden rule. Lot's of twists and emotional turns, very good show animation, except for one part to date where they used 3d and it looked god awful. Otherwise, great show. To be honest, 100x better than the previous version of the show, better story.
(TENGEN TOPPA) Gurren Lagann: I know you said you didn't like the action shows, but this one really is the grand-daddy and really represents modern anime. It's very comedic and is a prime example of some of the greatest animation to day in japanese anime. Every scene is hand drawn and there's no shortcuts evident in the series, done in a very traditional anime manner with some digital touch ups.
Click here to watch with subs (best link I could find). Click here to watch keyframe uploads via Gainax.co.jp It's in japanese, but on that page if you click on the pictures a flash window will come up with the the hand drawn frames from scenes of the anime. 27 Episodes, and 2 movie adaptations that retell the story in 1:30 minute format. I would watch the series instead of the movie, but the movie has some interesting new scenes that look great. There's also a lot of religious / scientific / sociological undertones in this show, and many political ones are aparent in the second half of the show.
Cowboy Bebop: Definitely a recommend, Watanabe is a genius.
Samurai Champloo: Definitely a recommend, Watanabe is a genius.
Paranoia Agent: Probably the greatest psychological anime to date. No action, pure dialogue and thriller events.
Detective Conan: It's fun due to it's murder mystery elements. I can't recommend it necessarily for it's story seeing as even after 15 years it's still going, but it's really some general fun in the form of murder mysteries, and it can be very clever at times. It's the only show I could ever reccomend that is over 50 episodes.
Eureka 7: The drama makes me cringe sometimes, but other times it brings tears to my eyes. Especially the ending. English voices are trash, don't bother.
FLCL: Another gainax show, ofcourse it's a recommend!