I will toss in the required non-Pixar animated suggestions.
The Wings of Honnamise- an incredible tale of a nations first attempt at space flight (its beauty and detail all the more amazing because it was created by a bunch of geeks after a fan film got some attention). Watch it subtitled, as the English dub changes two critical lines of dialog
Voices of a Distant Star- one man on a Mac created this incredibly moving short
Nausicca of the Vally of the Wind- actually, any Miyazaki film will do, but this is one of his best.
Watership Down- adaptation of the novel.
Batman The Animated Series- what can you do when told to make a kids superhero show? This. Also the anime series Mobile Suit Gundam for the same "they turned a toy commercial into THAT?" vibe.
LOL I double post.
I have to address this, cause I love this post.
1) Wings of Honnamise is a great film and is criminally unknown even in anime circles. Love that movie and how it handles the themes going on.
2) The same guy who did Voices of a Distant Star ended up starting his own studio, hiring some people and making a couple more movies (and probably more since it's been several years since the last that I'm aware of). First was "5cm/sec" which is a collection of three shorts about a guy's long-distance relationship with a girl at three different points in his life. I completely love the first of the three and consider it the best thing he's ever done. The second one is pretty good as well. The third is... not...
He then made a movie called "The Place Promised In Our Early Days" which is a feature film and is pretty damn good despite the way it uses the standard anime trope of going completely batshit insane in the last act.
3) Yes. Though Porco Rosso is my favorite of his movies, the manga of Nausicaa is one of the greatest fantasy stories ever told. The movie version is only about half the story told in a condensed way. I'd love to see someone actually do the whole series as a film trilogy or something. It is the definition of epic and I highly recommend the books.
4) Plague Dogs, by the same people who made Watership Down, is also quite good. I think Watership goes back and forth between being a silly kid's film and being a horrifyingly dark adult film. Plague Dogs is 100% pitch black adult film that only occasionally does something lighthearted or whimsical.
5) I have no comment on Batman.
I will make recommendation for you! If you've not seen them, check out:
1) Perfect Blue. "It's what Hitchcock and Walt Disney would make if they worked together" says idiot US film critic! It's a Kon film, so you've probably already seen it.
2) Now and Then, Here and There. Miniseries of - I think - 12 episodes, each an hour or so long. It starts out kinda goofy, with a kid who's sorta happy-go-lucky who meets a weird girl named "La La Lu" and then people riding giant robotic dragon pokemons show up to kidnap her. However, after a couple of episodes you realize where it's going, and then you eventually get to some of the climactic moments of the series and just... It's one of the better examples of character development / arcs and overall themes I've seen in anime. DO NOT watch the dubbed version. Dear god they brutalized a few of the characters in the dub.
3) Paranoia Agent. Dozen episodes. Most of the episodes focus on a different character, though there's more overlap to the stories in the second half of the series. It tackles a lot of different subjects, but deals mostly with people who are unable to cope with some serious problem they have and how they avoid dealing with it. This one, again, veers into batshit crazy territory in the last couple of episodes, but it's worth a watch.
4) Tekkonkinkreet. Batshit pretty much all the way through it, but has a lot going on under all the crazy.
5) Mindgames. Batshit doesn't even come close to describing what this is.
Sorry, I just don't often talk about anime. I usually avoid it like the plague cause I've found most of it to be really dumb or too obviously trying to be smart or whatever, and then they end with some Deus Ex Machina metaphysical BS. I stopped watching them because I'd get all invested in the characters and story and everything would build and swell into a climactic moment, and then it would anticlimax and I'd get narrative blueballs.
Also, Persepolis was a pretty good foreign animated flick.