A couple I haven't seen mentioned yet that I'm currently digging are:

ergo proxy (brilliant animation and story)
claymore
hajime no ippo (great sports anime)

I cannot believe that only ONE PERSON in this thread has mentioned Akira! The Grand-daddy of all (?) anime!

If anyone here hasn't seen it, please make sure you do ;)

...

I enjoy most anime (was completely addicted to Evangelion when it came out), but I particularly love everything from Studio Ghilbi (http://www.onlineghibli.com/)

My favourite is probably: Laputa: Castle in the Sky (awesome), followed by Princess Mononoke & Nausicaa of the Valley of the Wind.

Spirited Away, Kiki, Howls are great as well. All the movies are quite long, and require concentration - but I love them :))

Here goes........ :

Mai Hime - I don't know why, but I really -REALLY- liked this anime. It even made me cry......... *blush*

Mai Otome - Didn't like this one as much as Mai Hime, but remains a decent watch.

Bubblegum Crisis - The original series, not 2040. Even though it has much of an 80's feeling -which I despise in general-, I simply loveeeee it. Oh yeah, and Priss' ,cult, haircut!!

Bubblegum Crisis 2040 - I fell in love with it the moment I saw the intro! Best opening credits, IMHO.

Magic Knight Rayearth (TV Series) - Thank god, they followed closely the original manga story. Ryuzaki Umi FTW!!

Gravitation - Too much of a gay drama, but that's what makes it unique. Didn't like the soundtrack, though.

SailorMoon - This is what got me started......I've watched it like a thousand times and will watch it another thousand, probably. I can't be objective with this one, it's SailorMoon. I still remember waking up early on Saturday and Sunday mornings to watch the new episode. Ah, those were the good times....!

If we are including manga then in no particular order:
Naruto
One Piece
GTO
Rurouni Kenshi
Saint Seiya
City Hunter
... (at least two dozen more that I don't remember)

Angelic Layer

Big O

Bubble Gum Crisis (Original)

Full Metal Panic

Ghost In The Shell (both movies and both TV Seasons)

Gundam (Wing, X, G, Turn-A, Seed/Destiny, and MS08th Team)

One Piece

Rurouni Kenshin

Transformers (c'mon now!)

Vision of Escaflowne

Currently watching/enjoying Claymore, Gigantic Formula, History's Strongest Disciple Kenichi, Magical Girl Lyrical Nanoha Striker S, and One Piece

I'm gonna have to recommend Claymore, if you guys liked Hellsing, it has a somewhat similar feel to it. :)

 http://en.wikipedia.org/wiki/Claymore_%28Manga%29

My Anime goes in this order from first to current

Dragonball Z

Ruroni Kenshin

Big O

And of course Naruto

*Sigh* Has NOBODY seen School Rumble?
THE funniest thing I've ever watched.
I'm a pretty big fan of Gundam Seed and somewhat of destiny, though the ending sucked balls. Most of the n00b animes are good I guess, Bleach, Naruto Inuyasha. Then there's stuff like Kanon, Lucky star, etc.

Wow I never expected this topic to be so big. Guess there were more anime watchers overlapping with the gamers here than I thought. Cool to see everyone contributing!

Once again some other favorites that people have mentioned:

Big O, I really enjoyed the art style and plot on this one

Evangelion, of course it's a classic. One of the deepest animes ever made. If you want something to really make you think, this is your show.

City Hunter, loved the plot.

Princess Mononoke, Howl's Moving Castle, Spirited away: Hayao Miyazaki is brilliant. It's as simple as that. If you haven't seen these movies, do so immediately.

Bubblegum Crisis, The classic cyberpunk anime. I'll always have good memories of this one.

Detective Conan, I've always been a sucker for a good mystery, and this show still keeps pumping them out. I can't believe how long it's gone on.

Miyazaki's works are automatically disqualified from this list, they're too good. ;)