I will give you all the anime I'm familiar with, and they've all been mentioned before (I think):
Durarara!!--Stupid name aside, a really charming tale of high schoolers in Tokyo as they slowly get embroiled in the criminal underworld and the paranormal. Great characters, great story.
Baccano!--A story of Depression-era gangsters and bootleggers and their conflicts with each other told in a non-linear way, leaving you to put the pieces together yourself. Also great characters and story.
Neon Genesis Evangelion--A socially awkward teen is forced to pilot a giant robot to fight "Angels": giant monsters bent on wiping out humanity. Despite how cliched that sounds, the story gets so much better as the series progresses and you get closer looks at the main characters' neuroses. Can't recommend this one enough, and the finale movie (End of Evangelion) and the remakes (Rebuild of Evangelion). EDIT: Wait, only four years? You've probably already seen this one. Watch it again, anyway.
Full Metal Alchemist: Brotherhood--Okay I haven't actually finished this one, but this one has the most traditionally framed story (a continuing epic told linearly from episode to episode) than the others. The story (guy's brother's soul is locked in a suit of armor due to an alchemical accident and he wants a way to fix it) hasn't really rocked as hard as the others but has enough hooks to keep you engaged and the fight scenes are quite good.
Phantom: Requiem for a Phantom--Guy gets press-ganged into becoming a hitman for a secret criminal organization. Incredibly grim for about the first half, but undergoes a drastic tonal shift about two-thirds through that's a bit jarring. Still, a mature anime that succeeds in being dark without feeling emo.
Durarara!!--Stupid name aside, a really charming tale of high schoolers in Tokyo as they slowly get embroiled in the criminal underworld and the paranormal. Great characters, great story.
Baccano!--A story of Depression-era gangsters and bootleggers and their conflicts with each other told in a non-linear way, leaving you to put the pieces together yourself. Also great characters and story.
Neon Genesis Evangelion--A socially awkward teen is forced to pilot a giant robot to fight "Angels": giant monsters bent on wiping out humanity. Despite how cliched that sounds, the story gets so much better as the series progresses and you get closer looks at the main characters' neuroses. Can't recommend this one enough, and the finale movie (End of Evangelion) and the remakes (Rebuild of Evangelion). EDIT: Wait, only four years? You've probably already seen this one. Watch it again, anyway.
Full Metal Alchemist: Brotherhood--Okay I haven't actually finished this one, but this one has the most traditionally framed story (a continuing epic told linearly from episode to episode) than the others. The story (guy's brother's soul is locked in a suit of armor due to an alchemical accident and he wants a way to fix it) hasn't really rocked as hard as the others but has enough hooks to keep you engaged and the fight scenes are quite good.
Phantom: Requiem for a Phantom--Guy gets press-ganged into becoming a hitman for a secret criminal organization. Incredibly grim for about the first half, but undergoes a drastic tonal shift about two-thirds through that's a bit jarring. Still, a mature anime that succeeds in being dark without feeling emo.