I really don't like the "big name" series. Naruto, one piece, pretty much anything that you hear about on a regular basis has never interested me...
The point of all this is I'm trying to find a new one to watch, and I don't know enough to find out which one is worth my time of looking into. My current favorites go as follows:
1. Serial Experiments: Lain
2. Elfen Lied
3. Gungrave
4. Last Exile
5. Ergo Proxy
Others: NGE remakes, Claymore, Berserk
The more philosophy and character depth, and preferably the darker the atmosphere, the more I'm going to like it. Do you guys have any suggestions?
Some of the responses here seem a bit weird to me, since it
sounds like OP's taste is "story-heavy and definitely not shonen", and some of the recommendations are exactly the opposite of that (and dear god why would you ever recommend SAO to anyone, especially someone that actually wants a good story). But anyway, some suggestions:
Death Note and Code Geass are obvious choices, though you've probably already watched those. If you haven't, go do that before you watch anything else!
Deadman Wonderland
Forgive the dubs. From the creators of Ergo Proxy, sounds like exactly what you're looking for.
Psycho-Pass
[C]: Control - The Money of Soul and Possibility
Follows the story of a kid who finds out about a secret underground society in which you can borrow money with "your future" held as collateral.
Mirai Nikki
Was already suggested, but I'll bring it up again, it's worth the mention.
If you're willing to go less action-oriented, other good story-heavy series:
Mushishi
http://www.youtube.com/watch?v=5PloHXm9e8E
One of the best anime I've ever watched. Make sure you watch it subbed, it's very important to preserve the overall feel of the show. Almost every episode is its own self-contained short story, so you can watch and much or as little of it as you like.
Zetsuen no Tempest
Only anime I've ever watched that actually spent 6 episodes on a single scene and I actually enjoyed and was at the edge of my seat for the entire thing. Takes a couple episodes for the story to pick up, but definitely worth a watch.
Spice and Wolf
Takes a few episodes to pickup. It's a bit hard to explain why this is good, the plot itself isn't that special on paper; the relationship between the characters (and random facts about economics and trade) is what really makes this shine.