MONSTER-Deep psychological thriller that deals with ethics and morality, and what it means to take and give a life
Baccano!-A fun and violent collection of multiple plot-lines and stories that connect and jump around through time in interesting and unpredictable ways. Set in 1920s America
Durarara!!-Set in the same universe as Baccano! and deals with the same type of storytelling-this time in modern Tokyo, Japan.
Fate/Zero-A prequel to Fate/Stay Night (but much darker/mature and better). The story is set around a magical battle-royale where seven mages summon spirits of legendary heroes in order to combat each other for the prize of the holy grail, which is said to grant any wish.
Kino's Journey-13 episode long anime that revolves around a girl named Kino travelling to different, fictional countries, with her talking motor-bike, Hermes.
Psycho-Pass-More newish anime. Something along the lines of Stand Alone Complex combined with Minority Report
Kara No Kyoukai-These are seven movies animated by the same company that did Fate/Zero. Non-linear, super-natural psychological thriller. Very gorgeous, but can be a bit of mindfuck and you need to pay close attention, since movies jump around non-chronologically within the time-line of the story.
MushiShi-Similar to Kino's Journey. It revolves around Ginko, a mushi master, who goes to different towns trying to solve different people's problems with the mushi, which are these mythical creatures that only certain, specific people can see.
Raxephon-Somebody called this an EVA clone, but I think the similarities are just superficial. Beautiful story, but drenched in symbolism, so prepare to be mind-fucked.
Out of all of these, I recommend Fate/Zero the most, which has recently gotten a dub (although I'd recommend the original japanese dub, even though the english one isn't bad) You can watch the entire series on Crunchyroll, in 1080p (and it's worth it, because the series looks fucking gorgeous)