I'm a person not given to having favourites of anything (except colours - purple, if you cared). I'll like some things more than others, certainly, but tend not to view one thing as being above every other related thing.
A brief mention of manga - I barely ever read it. As in, I've read about four or five manga series ever, and none within the past several years. So anything I talk about here, I'm talking about the anime version.
I've seen a lot of anime in my time, and forgotten most of it, so it's hard to make a wholly informed list of the ones that stand out for me. Perhaps that's just as well, though - if I've forgotten them, they can't have been outstanding, can they?
Tenchi Muyo was one of the first anime I ever watched, along with Dragonball Z (which I joined just before the Cell Games saga). While Dragonball Z was good and all, despite the hours of powering up and grunting at each other (I'm sure I'm not the first to point that out), it doesn't particularly stand out for me, other than in the "it was one of the first anime I ever watched" way. Tenchi Muyo, on the other hand, stands out much more. It was finite in length, had a comedy/harem plot (something I'm still drawn to nowadays), and still had magical powers and all that cool stuff.
After discovering anime, I looked around for the meagre pickings available on UK TV at the time. There wasn't much - some butchered Sailor Moon and Cardcaptor Sakura (both of which I have since watched in their original form and enjoyed, the latter moreso than the former), Neon Genesis Evangelion and Martian Successor Nadesico (both great stuff), and various films and series shown after midnight on the Sci Fi channel once a week (including Burn Up Excess, Urotsukidoji and various other things I don't remember - none of these were that memorable). There were a few other anime on TV at the time, including Cowboy Bebop, Outlaw Star, Inuyasha (I think), but I didn't watch them (I've since watched Bebop, but never the other two). Oh, and Pokémon. Yeah, that was fun, although I lost contact with it after a season or so. Digimon was also around, and I watched even less of that.
I then got access to Yu-Gi-Oh! (card games are serious business! I enjoyed it), Naruto, Bleach (both of which I'm still watching today - yes, even the fillers), a bit of .hack//SIGN, and a few other things.
It was when I went to Uni that I really opened up into the world of anime. I started the Anime Society with Fruits Basket, Paranoia Agent, Full Metal Panic and some other stuff, and I downloaded various things off the Interweb. From then on it's all a bit of a blur, so I'll end my reminiscences here.
In no particular order, here are some of the notable (to me) anime I've seen:
Bleach
Itazura na Kiss
Cowboy Bebop
Neon Genesis Evangelion
Martian Successor Nadesico
Cardcaptor Sakura
Gunbuster
Yakitate!! Japan
Tenchi Muyo!
Tenchi in Tokyo (surprisingly, I've only ever seen this once, back when it first aired on TV over 8 years ago)
Genshiken
Ouran High School Host Club
Samurai Champloo
Lovely Complex
FLCL
Akira
Ghost in the Shell (just the films)
Anything with Lupin the Third in it
Maria+holic
Dirty Pair (mainly the "Flash" version)
Eden of the East
Moyashimon
Potemayo
Sayonara Zetsubou Sensei
Skip Beat
Toradora!
Tokyo Godfathers
Kimi ni Todoke
Katanagatari
Quite a few, certainly, and I've kept a number of anime out of that list that I don't think quite belong in there (although they're still good themselves). I've also seen several of the "popular" anime (see below), but again I don't think they quite belong on the list above.
- Suzumiya Haruhi, Sailor Moon, Dragonball Z, Clannad, Kanon, Naruto, Azumanga Daioh, Lucky Star, K-ON!
Also, I'm never watching Grave of the Fireflies again. Once is entirely enough.