Ghost in the Shell Stand Alone Complex for its interesting and mature storylines
Black Lagoon for its unabashedly ridiculous 80s action movie style and great characters
Planetes because anime doesn't do "hard" science fiction ever, but even if they did this would still be one of the best ones. It's about blue collar workers in space who have to retrieve space debris.
Moyashimon because it's super cute and good if you want to learn about bacteria and microbes (try it!)
I have to put in an endorsement for Great Teacher Onizuka anime or manga, though the manga goes on longer and just gets better and better.
The Twelve Kingdoms - Girl gets whisked away to magical world... but good. Seriously, the best anime/light novel with this premise. Spectacular character development and world realization.
Giant Robo - Probably the greatest OVA ever made from back when OVA meant high production values. Took 10 years to finish, old school character design, epic storyline and animation is basically feature film quality.
Vagabond, Musashi Miyamoto story. Beautiful artwork and very engrossing.
Akira, movie covers two volumes, but then it goes on for 4 more volumes!
I'll reiterate support for Berserk here.
Full disclosure, parent company Media Blasters released Twelve Kingdoms and Giant Robo on DVD, but seriously they're freggin' awesome.