Nobody else has mentioned this one, so I will: Neon Genesis Evangelion. Starts out as a stereotypical "teenager-in-giant-mech-fighting-monsters" anime but it gets really good both plot/character wise and in some philosophical concepts they touch on. Its a relatively short series too, only 25 or 26 episodes. And if you are at all into esoteric religious philosophy, you will appreciate the symbolism and references throughout the series. (even without any knowledge of that you can still enjoy the series for what it is)

Without spoiling anything, the original ending was controversial enough that they made a movie to rectify it, (End of Evangelion), and there are a few other sequel movies based on the series.