This can be a daunting question, and my answer is always going to be subjective.
However, these are the ones I find most appealing and that I think best capture how well an animated series can handle a story with universal appeal.
- Akira
- Cowboy Bebop
- Neon Genesis Evangelion
- Spirited Away
- Studio Ghibli films (all of them!)
- The Melancholy of Haruhi Suzumiya [Series later adapted into animated movie]
- Natsume’s Book of Friends
- Code Geass: Lelouch of the Rebellion