Well there is no such thing as "best anime", since it's a pretty subjective matter.
I do have a few opinions on what rates as a "good anime" though, but whether it is the "best" is up to individual preferences.
In no particular order, anime that I think are good (or classic, whatever)
1) Serial Experiments Lain
- because it actually made my head hurt while trying to grasp all the concepts. one of the "thinking" anime types, you definitely have to watch it more than once to appreciate everything it has to offer.
2) Anything by Hayao Miyazaki
- cos he's a legend?
I loved Spirited Away.
3) Grave of the Fireflies
- a bit emo. but it IS a wartime story, so yeah. a touching one to watch. highly recommended.
4) Excel Saga
- this is only here because it's the most outrageously stupid and funny thing I've ever watched...and that's saying something. You'll find all sorts of off-the-wall references to pop culture and what-not. Not to mention finding just about every cliche in the book being used in every episode. Don't try to figure out what the heck is going on, just sit back and laugh at the sheer idiocy.
And did I mention that they break the 4th wall repeatedly and excessively right from the start of the first episode (and to the end of the series). Haha.
5) Noir
- This is here because I'm biased.
And because I have a weird fetish for girls with guns. And the soundtrack is win. Anything by Yuki Kajiura is. Haha. Not to mention that I like the whole feel of the anime.
6) Revolutionary Girl Utena
- As it title implies, it is pretty revolutionary in certain ways.
It challenges the typical female stereotypes and the main character is able to break free of the constraints of both male and female expectations that her role originally should have given her, and redefines herself as someone who is wholly "her", and deserving of being a hero in her own right. Besides, the series is chock full of symbolism and craziness. Definitely need more than one viewing to understand it.
7) Neon Genesis Evangelion
- Everyone above has already said a lot about how awesome it is. Let me just add a bit more. If you want to see really dysfunctional main characters with severe problems (emo Shinji, crazy Asuka--try telling me she's not disturbed, emotionless Rei...you name it, you've got it) try to save the world from even more crazy people (Gendou, I'm looking at you
), this is it. Besides, I love all the religious references...Angels! Adam! Lilith! Human Instrumentality! Ok, never mind, just go watch.
Maria-sama ga Miteru
- Alright alright, if it isn't obvious by now, I watch a lot of shoujo-based anime. Gah. But anyway, watch this if you like watching character interaction. Some people might not appreciate it because there isn't any hardcore kind of action, but people who like watching subtle interplay will definitely see what's the draw. (and of course the fact that I strongly suspect that the whole cast is gay has nothing to do with it. nothing at all)
Hmm I'll put more up when I think of them next time. Anyway, I think these are worth watching. Best? I'm not sure. They're still very good though.