Pretty much what has already been said, and then add:
Monster - A doctor searches for one of his former patients that has turned into the most horrible sociopath imaginable. The man has an unknown past and no name, and the story takes place throughout several years in Germany and the Czech Republic. Excellent if you like murder mystery/thriller/horror.
Tengen Toppa Gurren Lagann - A story of a boy growing up to be a man...a lot of mechas. This anime is not an ironic poke at 80's action or action in general. It is a celebration of it. This is the only anime that i know of where explosions explode.
Hunter X Hunter - An anime about a boy looking for his father, who is presumed lost. It turns out that the father was a member of an elite organization of "Hunters" and the boy wishes to become one himself. Best damn adventure feel that i ever got from any medium. Watch the older version for a better soundtrack and pacing.
Death Note - A kid finds a book that happens to belong to a shinigami (="death god"). It turns out that when you write a person's name in it, he or she will die shortly thereafter. The kid soon starts to exploit the power that the book has. Did i mention that the kid was a megalomaniac with a ridicolously high IQ and charms?