I know the thread title says to choose one best show, but I'm gonna use this as an excuse to gush sycophantically about my favorites like everyone else.
Mushi-shi - A really beautiful, creative show that never fails to make me sleepy. Slow-paced, but an incredible stab at (what feels like) a bunch of totally original Japanese folk-tales.
Mononoke - Shitty animation is cleverly disguised by some of the most eye-catching and colorful art direction ever devised. A really brilliant and abstract look at the frailty of the human heart that secretly wants to be a show about scary ghosts. Or is it the other way around?
Eve no Jikan - Androids and humans co-existing peacefully and learning about each other and shit. Very, very old ground for sci-fi, but this show manages to make it all seem fresh again by transplanting all the philosophical discussions to a Starbucks.
Wolf's Rain - Best ever soundtrack. Really, friggin' good English dub. Excellent animation. Religious subtext. Wolves who become friends. Go watch.
Kino's Journey - More philosophy stuff, but this time it's all pretty pessimistic. Basically a travelogue of a neutral observer who watches people in various fictional countries be horrible to each other. Also, there's a talking motorcycle.
Ergo Proxy - Is anime these days too bright and cheerful for you? Altogether not enough emo-gothic art design and shades of gray. Try Ergo Proxy! Now with more awesome game show episode!
Serial Experiments Lain - I've heard that a college professor made his students watch this in class. So yeah, if you like cyberpunk, watch this and people will think you're smart.
Monster - Yay, Monster! Fun fact: My avatar is from this show. Also, I challenge you to find a show of any medium that has a better villain.