I don't like concentrating "Anime" into a single genre, for two reasons: One, 'Anime' is just a contraction for 'animation' and in Japan it literally is used for any form of animation (One time when I was living in Japan, I saw a misguided Anime Nerd ask the clerk for the clerks favourite anime, and started whining when the clerk gave him a copy of 'Fantasia' and saying that it wasn't anime) and secondly, like movies and books there are tons of different genres. When people ask me "Do I like Anime" I always feel weird because there is a lot of anime I don't like and a lot of it are the shows that most people think of when you say "anime," yet there are some Japanese Animations that I absolutely love. For example, Princess Mononoke and Innocence (I refer to it just as Innocence because I didn't like the first Ghost in the Shell) are in my top 10 movies of all time list.
To make it short, I'm very picky but I don't like to squeeze anime into a single genre. I just judge them on their own merits. Regardless, here's a list of Japanese animations I like.
Innocence (One of the best sci-fi movies ever made)
Death Note (Very clever and creative show with good animation and a great story)
Naruto/Shippuden (Forgive me, I will agree it can be annoying and the 'really big' fans of the show are annoying, but I got hooked on this show early because I saw it when it first premiered in Japan as I was living in the country at the time.)
Trigun (I actually haven't seen it all, but I love the setting and the characters)
Ghost in the Shell: Stand Alone Complex (A cool show, though I lost track after the ending of the 'Complex' arch.)
Princess Mononoke (Epic mish-mash of Japanese history and fantasy)
Spirited Away (Beautiful and a ton of fun)
FLCL (Insane but hilarious. I don't care for the action sequences because I don't like giant robots, save for the one in episode 5, but it plays a fairly small part so I forgave it. Plus its hilarious.)
Dead Leaves (Don't go into this movie without an open mind or a seat belt, you might get whiplash and your brain might hurt from being fucked so hard)
Grave of the fireflies (This movie made me cry buckets :'( Good though)
Paprika (This movie is creative, gorgeous, and very entertaining)
Tokyo Godfathers (A very fun comedy drama from Satoshi Kon, who did Paprika as well. Satoshi Kon's animation is by far the best in the industry next to Hayao Miyazaki, its so much more expressive and full of motion compared to most Japanese animations)
Paranoia Agent (Another Satoshi Kon, but this time a TV series. Fucked up, but awesome.)
Mahoromatic (Forgive me again, or kill me, but I liked this show. I know it's full of bad fan service and I know it's what you can call a "Chick show" but my friend Emma from Australia got me hooked and it charmed the pants off of me, so :\)
Hellsing (Cool show. A bit misogynistic to the point of offensiveness at times, but still cool.)
I can't think of many more right now.