I am extremely picky when it comes to anime. There's more crap in anime than you can imagine. There is a lot of anime die-hards that will love whatever comes out of japan, making reviews and opinions hard to accept as unbiased.
Having said that, I'd like to say, I LOVE good anime. An anime done well rises beyond its genre as anime. The one that I fell in love with was also what started my love for the Cyberpunk genre; Ghost in the Shell. The original movie was a thinkers action flick. When you hear Movie Bob talk about movies that are deep but have enough flash and bang to keep most audiences engaged, you can include Ghost in the Shell in that mix. It has a lot of concepts of what is human, at what point are you still a person, and where can that dynamic expand to? Then on top of that is a nice layer of cyborgs, automatic gun fights, explosions and car chases. This movie I prefer watching in English. The voice actors do a pretty good job. Future inclusions to the name (there's a series that's seperate from the movies, but plays close to the manga) are almost like it's meant for english dub.
In the same era of anime, "Akira" is a classic. Also in the Punk genre. It's sci-fi, dystopian future, but has some of that super power stuff that animes are spotted with. Very tollerable doses of it though. Less "deep" than Ghost in the Shell, but still very enjoyable.
Ninja Scroll; same era as the above two. Imagine if you will, super powered ninjas with very iconic powers unique to themselves to one on one with the hero of the movie. Not in ANY way deep. No sir. This is pure violence, pure action, very enjoyable reguardless. It'd be like if Naruto grew a pair. Only, yaknow, good.