Hmm, let me pull out my list. (yes, I actually keep track of every anime I've seen and have them ordered in a text file)
1. Princess Mononoke -- Absolute must. Brilliant direction, unforgettable moments, intelligent story, handles environmental themes without being preachy. Hayao Miyazaki's best.
2. Spirited Away -- Also an absolute must. Miyazaki's second best.
3. Summer Wars -- This is by a relatively new director called Mamoru Hosoda, and I think in a few decades he's going to be nearly as beloved in Japan and among Western anime fans as Miyazaki is today. Summer Wars in particular has phenomenal characterization considering its immense cast, thrilling action scenes, and is profoundly touching.
4. Redline -- Break this one out for balls-to-the-wall testosterone night. Far and away the best example of traditionally animated vehicles.
5. Akira -- Kickass and thematically powerful. Good view into late 80's Japanese zeitgeist and also exciting as hell and exquisitely animated.
6. Grave of the Fireflies -- One of my math teachers in high school called this the saddest movie ever made, period. He was absolutely right. Best of non-Miyazaki Ghibli, and also important historically. Bring tissues.
7. Cowboy Bebop: The Movie -- The TV show is indispensable, and this movie is like a long, excellently animated episode. Notable for matching action perfectly with soundtrack (Yoko Kanno!).
8. Evangelion 2.0: You Can (Not) Advance -- This is part of a series of movie remakes of the Neon Genesis Evangelion TV series, and in my humble opinion the remakes are actually superior. It's powerful stuff, and there is no better mecha anime.
9. Kiki's Delivery Service -- One of Miyazaki's earlier movies. Heartwarming fun for the whole family.
10. The Girl Who Leapt Through Time -- Hosoda's first "real" movie. Emotional, charming, funny, and outstanding character animation.
Honorable Mentions:
- Anything directed by Hayao Miyazaki, ever.
- Ghost in the Shell (1995), if you're in to philosophy-heavy, action-packed cyberpunk.
- Hosoda's two Digimon movies are worth seeing. They were released in butchered form as Digimon: The Movie in North America along with one other short Digimon movie. I hate to say it, but watch these fansubbed. Look for "Digimon Adventure (movie)" and "Digimon Adventure: Our War Game."
Dishonorable mentions:
- I would not recommend seeing any of the Pokemon movies, except maybe the first, and even then that's a stretch. It's just not a good movie. (Don't hate, I'm here because of the games.)
- I don't care what any delusional old-timers say, Ninja Scroll is not required viewing. It's gratuitous and stupid. I couldn't sit through the whole thing. (Note that the old-timers are absolutely right about a number of other movies, such as Akira and Ghost in the Shell.)