I specially tend to like surrealist and deep stuff; realistic animes with mainly psychological and drama elements.
What I liked are things such as:

Paprika & Millenium actress (Overall, Satoshi Kon's works)
Serial Experiments Lain
Tenshi no Tamago
Kemonozume
Akira
Evangelion series
Mind Game
Cat Soup
The Sky Crawlers
Kaiba

Stein's gate

Akagi
One Outs

another satoshi kon
paranoia agent

Natsuyasumi.

mnemosyne

+1 for a cyberpunk fan

Blassreiter

Blue Gender

bubblegum crisis tokyo 2040

Ghost in the Shell:SAC

and ofc Cowboy Bebop and Outlaw Star

Not really realistic but great anime

One Outs
Kaiji
Welcome to the NHK

definitelies:
boogiepop phantom
angel's egg woops
tekkon kinkreet
texhnolyze
denpateki na kanojo

possiblies:
mouryou no hako
aoi bungaku
tamala 2010
connected

maybes:
time of eve
petite cossette
karas
bungaku shoujo
kakurenbo
pale cocoon

This post was edited by Xenon[KoA] on Mar 21 2013 12:31am

Tatamy Galaxy
Haibane Renmai
Revolutionary Girl Utena
Kino no Tabi
Ghost in the Shell (especially the first movie and the 2 TV shows)
Jin-Roh
Texhnolyze
Key the Metal Idol
Memories
NHK
petite cossette
Angel's Egg

Or maybe one of these I didn't like them so much but they fit your description:
Ergo Proxy
Genius Party
Rahxephon