Fate/Stay Night is a good one. The heroes from old legends are reincarnated to serve under masters as they fight each other in the search for the Holy Grail. One of the heroes is the main character from the future, who has come back in time in order to kill himself in the past.
Cowboy Bebop is one of my personal favorites. A space bounty hunter searches for bounties and cash in an attempt to forget his past (or maybe in order to seek it out). Everybody is out to get everybody; even his own team feels no guilt in cheating him, and vice versa. A very deep story.
I'm surprised
Yu Yu Hakusho hasn't been suggested yet; think Bleach crossed with DBZ. A boy is killed as he sacrifices himself to save a young child, and is given a second chance at life under the condition that he hunts down ghosts and demons from the other world.
Trigun is always good. A mysterious gunman lives by a "no kill" code. He's wanted for the destruction of an entire city; surprisingly, despite the fact that the city of July was destroyed, nobody within July was killed. He always acts aloof and carefree, but you quickly learn that he has many deep emotions hidden away.
InuYasha is good for the first couple seasons, but I lost interest after a little while. A young girl falls through a well and is transported to the distant past. She meets a half-demon who is bound to a tree by an arrow, and only she can remove it. When she does, the two set off to find the wish-granting jewel so that the half-breed can become a full fledged demon.
Rurouni Kenshin starts off kinda slow, but comes back hard in the second act. A samurai known across Japan as the "Man Slayer" vows never to kill again. He travels across the land, masterless, in a search for redemption. The sword he wields does have an edge, but it is on the wrong side; this allows him to use his sword normally without fear of accidentally killing. Despite his happy and calm exterior, he is constantly locked in a fierce struggle with his inner rage.
Outlaw Star is one of a kind. A young man dreams of going out into space. When he finally gets the chance to, he's greeted with the wrath of a crime empire and the legend of a great treasure. While doing his best to avoid the syndicate, the young man makes his way in search of the galactic mother load.
Ninja Nonsense is one of the funniest (if not THE funniest) anime I've ever seen in my entire life. A young girl is training to be a ninja. But her perverted classmates (who are all comically named Sasuke) and equally perverted teacher (voiced by Sean Schemmel who also voices Goku) are always causing her trouble, and her oblivious innocence only makes it worse. Her only hope is her "more than friend less than lover" friend, which seems to be the only level headed character in the entire show... It doesn't help that the young konuichi starts developing feelings for her friend (who is also a girl). If you enjoy comedy of the perverted variety, then this is a must see!
Another crazy comedy is
Sayonara, Zetsubou Sensai. It's like FLCL taken up a few notches. A teacher (who's name "Zetsubou" roughly translates to "depression") is depressed and is constantly trying to kill himself. However, this never works. So he comes into school each day and (instead of trying to empower his students with happy thoughts) works to motivate the class through harsh and depressing truths about the world (the "you will never succeed" variety). And of course, many of the girls in the class have a crush on the teacher (they are also all psychotic in some way or another). It's only been subbed (and even then only fan subs) but it's still a great one to check out if you like funny anime.