No one here mentioned MACROSS PLUS?? Get with it, people!
Seriously, the Macross universe is up there with the greatest film franchises in my book. Granted, there are a lot of sequels/prequels, but you don't need any previous knowledge of the story to see PLUS, and it delivers.
And, of course, the original Super Dimension Fortress Macross (which was force-fit as the first part of Robotech), may be dated in animation, but the story is pure magic to this day. It has been released uncut by AnimEigo and ADV, so you can see all the stuff they mangled/edited out for the Robotech version. And if you ever want to be blown away by the finest hand-drawn animation ever conceived in the 80s (
to this day, I'd almost argue), then the film version, Do You Remember Love? is an absolute anime CLASSIC! That's the one that solidified my love of the genre back in the 80's.
The newest series in the franchise, Macross Frontier, is also a great place to start, but it hasn't been released on dvd yet, as it is currently airing in Japan. You can download fansubs weekly, though.
A lot of people here mentioned the more obvious ones, like Akira, Ghost in the Shell TV and Cowboy Bebop, but definitely check these out if you get a chance:
Gurren Lagann (though you may want to wait until you've seen more of the genre first, since you'll probably never find another series as mindblowing and EPIC!!!)
Bubblegum Crisis (the original 8-part OVA series, NOT the "Tokyo 2040" tv series. Fear not, the name of the show is very misleading. It's like Blade Runner on speed. I actually think a LOT of people here that are into Terminator, Blade Runner, etc. would go crazy for this if they gave it a chance. But avoid the English dub like the plague, as it is truly AWFUL.)
Escaflowne (Knights battling each other in giant robot suits of armor in an incredible fantasy world. What's not to like??)
Just about anything by Miyazaki, but particularly Nausicaä.