Not much of actual obsessions, but after watching Sketchbook ~full color'S~ I went back to drawing and painting after a almost a year long break... Considering how important that part of my life is, I wonder where would I be now if I had abandoned art completely.
I'm pretty sure that if I saw Bamboo Blade when I was younger, I'd start learning Kendo - it was the only sports anime that ever made me interested in the actual sport.
One thing I'm pretty sure I picked up from anime(or rather manga in this case) is my love for tea. I always liked tea, but it was mostly the instant, cheap kind. After my fascination with Japanese culture began to grow, I started drinking Japanese green tea. And some time ago, inspired by few rather girly manga(some shoujo and yuri) I got interested in good quality "western" tea, and by now I have a hard time drinking the instant stuff.
Oh, and one more thing I got from those manga: I always loved to cook, but they made me interested in making sweets - cakes, cookies, and so on.
And of course there's the huge fascination with Japan itself - I was always a bit fascinated by contemporary Japanese culture, but anime and manga made this much more pronounced. Especially the Japanese language - I'm learning it not because I want to read manga and watch anime in the native language(that would be a bit silly), but because it's the most beautiful language I have heard any significant amount of in my life. Other then my native language, I learned English, some German, some French, I also heard my fair share of Russian and Arabic(the last one thanks to my grandmother, who spent some time working in Libya as a nurse), but none of those ever fascinated me as much as Japanese.
By no means do I think that Japan is a perfect country - but the things I like about it outweight the things that turn me off, and that's a rarity in the current world.