Heh. That's an interesting question to say the least
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Well. A horribly translated anime was my introduction, but, thankfully (?) for the future of my liking of anime, it was a very...hot episode
What first got me into anime was watching chicks in a hotspring

It was the DiC translated Sailor Moon. I had previous experiance with DiC shows, like the original Megaman and a few others. I think there was this lizardman show...or something... Anyway... You can blame early puberty on my initial interest
Eventually, that evolved. As I matured, my interests changed. Sailor Moon still holds a special place in my heart, especially after seeing the original, well-translated anime and reading the manga. However, the
reason for watching anime evolved, even as watching DiC's Sailor Moon. I realized that there was story and adventure behind it.
I love a good story. There are excellent comedies for a laugh between drama. Excel Saga is a good betweener for the semi-seriouness of Trigun and VanDread and the pure suspense of Ghost in the Shell. You have such a wide-range of story-types and literary works that the selection is definately out there.
What I find intriguing, more than anything else, is the seemless blend of culture presented in many anime I watch, especially the science-fiction/fantasy. You have shows like Slayers and Those Who Hunt Elves that mix and match medievil culture with fuedal. You have shows like Macross/Robotech and VanDread that seemlessly blend American, European, and Asian culture in a believable sense.
There's always personal development, and
stable development, of each and every single character presented. You know the character, and you almost always see how the character learns from their experiances to become somebody...else. You have intimate stories, and more than just main/primary characters have evolutions and epiphanies.