Irresponsible Captain Tylor. Funniest series ever made, and the best ending to a series I have ever seen.

Tenchi Muyo! Watch the OVA. The movies are wonderful as well. Avoid Tenchi in Tokyo cause it was made by people who hate children and puppies.

Rahxephon, It's like Eva for the first half of series and then evolves into it's own monster of a headcase.

Wolf's Rain. Cause it's beautiful.

Like Renmiri, anything by Hayao Miyazaki.

Gungrave, mafia shennanigans, betrayal, violence, drama, and ...zombies

Naruto, I know people don't care cause it's "mainstream" but find a fansub copy of the series (the fansub was kind enough not to translate Naruto's obnoxious catch phrase) It's a very deep an well written story about growing up, being accepted, honor, friendship and believing in your dreams. Even better, it's actually well written and doesn't feel too forced most of the time.

Robotech or Macross (depending on how much of a purist you are) The world is saved by a idol singer...

Night on the Galactic Railroad, a beautiful tale about growing up, life, death, and learning to let go.

Captain Herlock or Galaxy Express. Cause Leiji Matsumoto is a genius.

That was more than five but I have like thirty other series I could recommend like Sailor Moon, Ranma 1/2, Casshan the Robot Hunter, Vampire Hunter D, Robot Carnival, Kare Kano...