WARNING: Most anime is incredibly stupid, created for the sole purpose of selling toys and video games, or panders to the basest and creepiest aspects of the otaku subculture. Consume at your own risk.
That said, the "Paranoia Agent," "Monster," and "Cowboy Bebop" recommendations are right on. Also check out "Serial Experiments Lain," "Haibane Renme," "Ghost in the Shell" (the first movie was innovative for its time and is still awesomely entertaining, the second movie is Mamoru Oshii's "eff you" to general audiences by way of its comparative emphasis on philosophy and lack of action, and the "Stand Alone Complex" TV series is a fun police procedural that makes those of us born during the cyberpunk sweetspot remember when it was a viable genre), "Kino's Journey," "Eden of the East," "The Girl Who Leapt Through Time," "Grave of the Fireflies," and the second "Patlabor" movie. The first season of "Gunslinger Girl" is spectacularly animated and thought-provoking sci-fi that subverts typical moe/lolicon otaku bait, but the fact that it has to employ the genre tropes it deconstructs and criticizes to effect its critique makes some viewers uncomfortable.
The bad news, though, is that "Evangelion" is basically the medium at its finest. The recommendations you're getting in this thread are good, but I doubt any of them will move you like Eva did. Did your Eva immersion include "End of Evangelion?" If not, run out and buy a used copy ASAP. It's the feature-length capstone the series deserved but couldn't afford on a television budget.
