Fine, I'll run down a quick top 10 off the top of my head. Also note that some of the series below you can find online through Crunchyroll or Hulu, legally, for free. In no particular order...
1) Darker Than Black
Chinese Electric Batman fights against "Contractors." They're kind of like X-Men mutants except that they all have a compulsion to pay a certain "price" every time they use their powers. The price can range from smoking a cigarette to breaking your own fingers, but the powers are pretty amazing. Animated by Studio Bones and with a musical score composed by the great Yoko Kanno. You will like this show, though the second season is a bit of a letdown.
Probably the closest thing Japan has to The Simpsons
. A samurai-in-training meets up with a lazy ex-samurai and a strong alien girl voiced by Rie Kugumiya and they go on lots of crazy adventures that often reference Japanese pop-culture and other anime/manga franchises. Yeah, the manga has gone through hundreds of chapters and the anime has likewise hit 250-something episodes but you can jump in at nearly any point and not be too lost because the story rarely has any continuity aside from some mini-arcs.
Also, doesn't this song
just bring a tear to your eye?
3) Tengen Toppa Gurren Lagann
Yep, gotta put it here. A mech show that rarely took itself too seriously, so that the few times it did it really felt like an effective sucker punch to the gut. Plenty of BURNING SPIRIT and DRILL POWER to go around multiple times. I would watch this show over and over again.
4) Cowboy Bebop
I cannot think of a single episode I wouldn't watch multiple times. Amazing soundtrack, characters, soundtrack, dialogue, soundtrack, action scenes, and did I mention the fucking soundtrack that was composed by Yoko Kanno? Because I don't think I did. You cannot call yourself an anime fan without watching this whole series at least once. Also, I actually preferred the English-dub voices to the original Japanese in this one.
5) Puella Magi Madoka Magicka
Not something you want to watch if you want to feel happy or still have faith in the goodness of humanity. But if you can get past that detail this show was amazing how Studio Shaft and Akiyuki Shinbo took the tropes of anime magical girl shows, twisted them, warped them through a funhouse mirror, saved a picture of that mirror's reflection, printed out a copy of that picture, tore that picture into strips and then glued those strips to their face before running out of the room screaming "IIIIIIIIMMA PRETTY PRINCESSSSSSSSS!"
Okay, maybe it wasn't that
crazy, but the way it just went deeper and deeper down the rabbit hole of what it means to be a magical girl
and ended with an ultimate sacrifice really warped my perception of this genre. Then again, I'm not sure if there will be another quite like this show.
6) Ghost in the Shell: Stand Alone Complex
Future-Japan, super-advanced robotics and cybernetics, transhumanist issues, no pointless fanservice, great animation, cool fight scenes and a female lead who isn't useless. Also, Yoko-fucking-Kanno composing the music again.
Based off a light novel that will never be translated over here (scattershot fan-translations notwithstanding), but even before I heard that it was adapted I loved this show. It's the closest we'll ever get to a Guy Ritchie anime, with a bunch of different mobsters with various quirks all stuck on one train for different reasons in Prohibition-era America. The English dub actually fits in this setting, and the VA behind Ladd Russo really makes him funny and endearing, if you can get past the character being a murderous psychopath.
I'll also jam Durarara!!
in this entry, because it's pretty much the same thing except set in modern-day Ikebukuro, written by the same author, and there's even a tiny bit of crossover with a couple of odd characters from the former appearing as cameos in the latter.
8) The Vision of Escaflowne
Old-fashioned knights mixed with weird mechs, awesome sword fights, a female lead who isn't useless and a villain who may or may not be Isaac Newton. Shame that the ending seemed a bit rushed, but otherwise I would totally recommend this one to anyone.
9) Last Exile
A world full of steampunk airships and characters who actually grow the fuck up in the heat of battle, though I still have no idea exactly who survived in the end (apparently everyone except my favorite character iirc). Still, I really love this show and have watched it more than once.
10) Black Lagoon
Adapted from Rei Hiroe's manga (and will hopefully continue if someone can drag him away from his femdom porno long enough to keep drawing it), kind of an homage to various exploitation movies. As long as you don't care about realistic physics it is an amazing show. Hell, it starts off when the good guys use an old Vietnam-era PT boat to ramp up into the air and destroy a helicopter before they hit the water.
Wow this list got bigger than I thought it would. Hope someone actually reads this stuff.