Trigun

Give this one the benefit of the doubt. It starts off as an apparently silly anime... but gets amazingly deep. While humanity escapes its dying planet, two children appear on the exodus ship. Upon crashing on a new planet, the story line follows the good twin, and questions morality, the caring for others, good vs evil... like the wiki says, it gets very emotional.

Trigun on Wikipedia

Black Lagoon

Similar setup as Cowboy Bebop, a group of people working as mercenaries. Set in tropical Asia, with Russian mafia and local lowlives. The episode with the little twins was really touching.

Love the opening soundtrack, even though it's a bit too "Engrish" for absolute enjoyment..

Black Lagoon on Wikipedia

Cowboy Bebop

If I had to pick only one anime series to have in my library, this one would be it. Set in the Space Western genre, it's a combination of great animation, awesome soundtrack (Yoko Kanno glides through every genre and creates a great atmosphere every time), cool characters, mixing human desperation and comedy, computer hacking, and away-from-Earth adventures.

Wikipedia Article

discovering anime

Anime explores every aspect of human life. It covers all genres, it fantasizes about every possibility. Some have cruder animation, but deeper meaning. Some are works of art, combining beauty and message.

This is dedicated to all those animes I've enjoyed the most.