Cowboy Bebop
Samurai Champloo
Desert Punk
Bubblegum Crisis
Blood the Last Vampire

A couple oldies but goodies - Vampire Hunter D and the Blue Seed series.

I'm going to pretend that the request for recommendation is from/for someone not very 'up' on anime already.

My two recommendations are the already mentioned Ninja Scroll and the thus far ignored Ghost in the Shell movie.

These two do not, naturally, stake out the limits of the genre by any means, but they are exemplars of what can be done in Anime in terms of writing and presentation, yet while both are decidedly mature in theme and presentation (as compared to juvenile, not a more puerile mention), they are also not at all similar viewing experiences.  Akira comes in a not too distant third.

Of course, none of those are reasonably contemporary.  Obviously Miyazaki's canon, as mentioned, is excellent and fresh, though may not appeal to everyone.  I personally watch Bleach, but wouldn't exactly recommend it as a starter unless I knew the tastes of the person I was recommending it to, as the quality of the writing grows progressively worse as the series progresses. On the other hand, I actually wept (true, they were tears of black tar, as my withered soul can not produce real emotion...) at the end of Bleach:Memories of No One, though getting there was hardly smooth.

I can recommend Gasaraki for the military/giant robot fans, a lighter, better exploration of character and family obligations than Neon Genesis Evangelion, which can be worth watching if you have a high tolerance for frustration, and also to fully enjoy the Shinji/40k fanfic epic that some wizard summoned up from a dream and posted upon the intertubez to enlighten mankind.

For the fantasy fan, Berserk was excellent, though strangely truncated.

For some reason, however, most of my favorites are all well over a decade old.  Still, here is a list of some other worthies which can be enjoyed variously depending upon your personal tastes:

Martial Successor Nadesico: Sci-Fi comedy/military drama Series
Demon City Shinjuku: Movie, supernatural police action/romance
Blue Seed: Series, postapocalyptic Giant Robot vs Monster
Fushigiyugi: LONG young teen relationship drama in a fantasy world series
La Blue Girl: Ninja/tentacle porn, but surprisingly funny and clever...for porn.
Blood: the Last Vampire: Short Movie (sorry, haven't watched the series yet...), schoolgirl/police/monster mashup set in 1950's Okinawa for whatever

I'll leave it at that as I could go on all day as my memory slowly boots up... and that's trying to avoid things already referenced here as well.

I don't generally like anime, and most turns me off right after the first episode/minutes, but loved Paprika for the surreal whimsy and lovely animation, the Evangelion TV series for its unflinching look at alienation, superficial human relations and clinical depression, Akira for everything (although the comic series was even greater with the Greater Tokyo Empire and all) and, well, I think that was it.

Vampire Hunter D and VHD Bloodlust are both good.
Cowboy Bebop is the best anime ever made.
Outlaw Star is excellent.
Witch Hunter Robin is good, though slow at times.

I second the vote for Akira, and also share the opinion that the GN is even more awesome. I also enjoyed Metropolis, the screen play for whaich was written by Otomo (the guy who did Akira).
As far as Miyazaki goes, I like his moves well enough, but in my opinion none of his work compare to the Nausicaä GN.

As the above might indicate, I find that I don't enjoy whole genres as much as I do the work of certain individuals working within them.

As it should be.

(And I have tried to get into the Nausicaa comics - got to the end of the first volume, but couldn't be bothered to ask for the second, which I took as a sign of "good but not for me").

Strangely enough, Melan, the first time I read it, I couldn't find a copy of volume 1 so I just started with volume 2*, which has a lot more cool shit in it. I like vol 1, but it might just be in retrospect.


*When I was a kid, living in Farmfuck, USA, finding comics in sequence was a rare thing, and I've never grown out of just reading what I could find regardless of the number on the cover.

It's short and getting oldish, but Voices of a Distant Star is fantastic.

So is Tokyo Godfathers.

EDIT:
In print, check out Tekkonkinkreet (AKA Black & White).

SECOND EDIT:
Oh yeah, and Osamu Tezuka's Buddha series is top-notch.