I was going to say "thinking man's" anime, but that would be sexist in leaving out the women. So let's go with cerebral -- an anime which either makes you think or requires you to think.
My choices:
Ergo Proxy. Re'l is a strong, independent woman -- or so she thinks. But take away her autorev and put her outside civilization, she really is close to hopeless. She can't even do her own hair. It takes a monster and a robot to teach her about humanity. When Pino asks, "Is Pino Pino?," it's a line that can take you to so many levels. What is humanity? Is it really reserved for the human race? Think about it.
Casshern Sins. Check out my avatar. Earth may be dominated by robotic droids, but their time is limited after Casshern kills Luna. Why does he kill her? That's what Casshern wants to know. And is Luna really dead? With all these robotic warriors, there is action, but mostly you again ponder the basics of existence and the purpose of life. Little Ringo seems to understand. She knows she is doomed, but still sees the beauty of the world.
Grave of the Fireflies. The most depressing anime film I have ever seen, but don't let that scare you off. It really makes you ponder the cost of war. This one will not leave your mind.
Now your turn.
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