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I'm gonna toss in some oddballs here. I watch a pretty odd swath of animes and these are the ones that really stuck with me. Let me get the obvious ones out of the way first though.
Bebop/FLCL/Evangelion
and now for the oddities.
Gunbuster+Diebuster (The full OVA's. Not the movies)- Straight from the same studio that did TTGL, it's just as crazy and epic. Moreso even.
Planetes- One of the best slice-of-lifes you'll ever watch. And it's in space.
Kaiba- This is definitely for people who love the Fiction part of Science Fiction.
Dennou Coil- Google Ghibli Glasses (not actually Ghibli but it has a similar vibe)
Paranoia Agent- A classic. Good for mysteries and brutal reality.
Samurai Champloo- Edo Era Hip Hop Fusion from the mastermind behind Cowboy Bebop.
Kino's Journey- I'm not gonna describe this one. Look it up. Or better yet, just watch it without any expectations.
The Tatami Galaxy- Extremely interesting and thought-provoking after it's conclusion.
The Big O- These mechs are not TTGL fare. They're more like godzilla. And they're glorious.
Cromartie High School- I have this song stuck in my head...
Ghost in the Shell- CYBERPUNKCYBERPUNKCYBERPUNKATTEMPTEDPHILOSOPHY CYBERPUNK