Since it seems as though everyone else has all the standards covered, allow me to suggest a couple that haven't been listed:
Rurouni Kenshin - If you're into samurai-period swordplay, few other shows can compete. It has ever been one of my favorite all-time animes with a great story, a believable "difficulty curve" for the hero throughout the story. And it has a great soundtrack. Also, if you can watch the three OVA movies (Samurai X) without crying at the last one then you have no soul.
Bobobo-bobo-Bobobo - Probably the funniest show - anime or otherwise - that I've ever seen, though this one I would strongly suggest trying to find the manga of. The show is still pretty damn hilarious, but the manga is an out-right riot. But I'll be the first to admit it takes a certain type of sense of humor in order to find this to be funny. It's absolute slapstick, 100% random and nonsensical...seriously, NOTHING makes sense, but that's how it's supposed to be, for it is the power of such insanity that gives the main character of the series his strength.
I've seen some other people suggest Evangelion, so allow me to slide this joke in. This is how a friend of mine described Neon Genesis when I was first thinking about watching it: "If you need therapy and you watch Evangelion, you won't need therapy by the end of it. If you don't need therapy and you watch it, you most certainly will by the end of it." Seriously, the last two episodes make less sense than the entire Bobobo series combined. "I am the me that I see myself as because I am me, not the me that other people believe me to be because their perception of me isn't the me that I truly am. So all I should strive for is to be the best me that I know how to be because I'm the me that I want to be." Orrrrr something like that.
Other mentions: While the Ghost in the Shell movie is great, the series that were made out of it "Stand Alone Complex" and "Second Gig" are very good cyber-punk series.
Lets see.....Cowboy Bebop's already been mentioned a thousand times, but it really is one of the gold standards of all of anime.
Think that about covers it for me, seeing as how everything else worth mentioning has already been mentioned.