I'd suggest the following if you want deep stuff.
"Ghost Hound"
"Serial Experiments Lain"
"Death Note" (not a fan myself but I acknowledge it's fandom and significance)
"Welcome to the NHK" (a personal favorite in this area)
Anything by the late, great Satoshi Kon.
For a more balanced approach to heavy stuff with more action and character development that touches on heavy themes though out the show but doesn't fixate on them I'd reccomend the following.
Old School Gundam (especially Zeta). If you see CGI robots and huge bright eyes walk away for now.
"Glass Fleet"
"Last Exile" (awesome as sin)
"Gunslinger Girl"
"Trigun"
"Rurouni Kenshin"
"Heroic Age"
"XXXolic"
Now if you're into slice of life there are some out there (believe it or not) that are more than just moe blobs and fan service. These ones are more cerebral or emotionally engaging and can be enjoyed by people who aren't looking for a 25 minute commercial for perverted, 1/6th scale PVC figurines of 12 year old girls in bikinis.
"Welcome to the NHK" actually fits in here kinda well.
"Kanon"
"Clannad" and "Clannad After Story" (these will make you cry so much)
From what I've heard from friends "Honey and Clover" seems alright.
"Princess Jellyfish"
Now luckily you can't watch most anime of the action verity with some sort of psudo-philosophical or psudo-intellectual message or commentary about love, war, morality, zealots or some other bit on the human condition, even if it's half assed and halfhearted, so I'd say if you have netflix feel free to look around. "Vandread" at first seems like a cheesy super robot and fan service shot but it actually has a pretty cool message that comes with it, as does "Gurren Lagann". I mean the end of action anime read like a self help book: "pick yourself up and control your destiny" and all that jive then some one gets rocket punched, pretty basic stuff, but it's usually there at some level or another. That's the difference between American TV and anime: American TV doesn't take sides (except against the terrorists because America) in order to not offend people and have a larger base demographic while Anime will flat out tell you that war is messed up, God doesn't give a crap about you and you have to do it yourself or some other stuff crap that for some reason is considered controversial around here.