Some great anime that I heartily recommend-
Gankutsuou: A beautiful, visually stunning retelling of Alexandre Dumas's classic "The Count of Monte Cristo", as told from young Albert's perspective. The story is well-told if slowly paced, and it's suspenseful enough to keep you wondering what the characters are going to do next, especially what the mysterious count's next move is going to be. The costume and interior designs contain all kinds of bizarre, shiny patterns that are fun to watch as they move across the screen.
Wolf's Rain: A sad and fantastic tale of four of the earth's last surviving wolves (who can change into human form at will) on a quest to find Paradise. Wonderful animation and a memorable soundtrack make this one a must.
Fullmetal Alchemist: I think most people familiar with anime know this one, but just to summarize, it's about the Elric brothers, a pair of young alchemists who make the stupid mistake of reviving their dead mother and have to pay the price for it- Edward loses an arm and a leg, and Al loses his entire body.
They spend the rest of the series on a quest to find the philosopher's stone, which can restore their original bodies, as they literally battle the Seven deadly sins. I've only seen the first few episodes, but I'm aware that it gets a lot darker and more complicated as the series goes on. It's very engrossing and emotional, and deserves a place on most fans' tops list.
Cowboy Bebop: I already mentioned that one, so need to go over it again.
Samurai Champloo: An excellent anime in a similar vein as Cowboy Bebop (and from the same director), about a wandering ronin and a slacker swordsman who have promised to help a young girl named Fuu find a samurai who smells of sunflowers, and once they are done with their quest, to duel each other to the death. Samurai Champloo boasts shiny, nice to look at animation, and a great soundtrack that boasts all kinds of different music (tho' most notably hip-hop).
There's enough humor to keep pretty much anyone laughing and there are enough anachronisms to give anyone a pleasant sense of dissociation from reality.
Loveless: Complex and brutally disturbing, Loveless is about a sixth-grader named Ritsuka whose brother was mysteriously and brutally murdered. Ritsuka meets a handsome young man named Soubi, who shows strong hints of being a pedophile and harboring a perverted attraction towards Ritsuka. Soubi claims to have been a friend of Ritsuka's brother, and offers to help Ritsuka solve the mystery of his brother's murder. Very good, but definitely not for all tastes.
Serial Experiments Lain- This series is confusing. Even after repeat viewings I wasn't able to figure it out. The series follows a lonely young girl named Lain who feels more connected to the internet than to real life. The soundtrack and art were both like something out of a really bad nightmare.
Outlaw Star- Very funny sci-fi anime. Think of it as an overdramatic family sitcom version of the original Star Trek series . Even the more serious episodes (not that this show has all that many of them) can make a person laugh
All of Hayao Miyazaki's films: Since Miyazaki's films are aimed at family audiences (especially younger people), they tend to be a bit preachy about the issues they deal with, such as anger management, protecting the environment, &tc., but they all have complex storylines and likeable characters.