I'm on other even though Bebop is on the list. You can't really say what is "best" because opinions very from genre to genre. But I'll put in some of my personal favorites.
Haibane Renmei
Serial Experiments Lain
NieA_7
Trigun
ANY Miyazaki movie (But especially Naussica)
X
Get Backers
Prince of Tennis (pretty good, but some might not like it, but I like sports anime so
)
Initial D (again with the sports anime)
Last Exile
Scryed
And just for fun...
Riding Bean (1980s like whoah)
Sakigake! Cromartie High School
EDIT
BTW, for those of you who don't know, anime does NOT mean japanese animation. It simply means animation. Borrowed from the english word animate, it was contracted to anime (ah-nee-may). In Japanese it would be improper to say anime when refering to japanese animation. You'd have to say nihon no anime (Japanese animation) or amerika no anime (american animation), etc. Anime simply means animation.
/rant of a student of the language
-Karrde-