I'm not a massive fan of anime, and I think it all stems from the first two I watched - Ghost in the Shell, and Urotsukidoji : Legend of the Overfiend.
Ghost in the Shell is, without a doubt, a brilliant film. It's deep, provocative and provides a great deal of insight into the human condition, at least from the minds of the writers. That didn't stop me thinking "The guys all have hoodies that turn them invisible... Why does Kusanagi have to get naked? What the fuck? (assuming this was an issue with anime, I tried manga, and was greeted with a page that actually had the words "FAN SERVICE" on it, and an upskirt shot of one of the characters...)
Urotsukidoji was... Well, the only time someone wasn't getting raped, being penetrated by tentacles or having vaguely penis-shaped tentacles spew "slime" was the menu screen.
I later tried to watch Neon Genesis Evangelion. The first 12 episodes are alright! Then... It gets all philosophical and is less about giant robots blowing up aliens. By Death and Rebirth I had no idea what was going on, even after repeated viewings, people explaining it to me and trips to wikipedia. I still haven't seen End of Evangelion, because I'd like to know what's happening before I go in.
Lastly, I tried Akira. This was alright! I was eventually put off by the guy I was watching it getting all grumpy and huffing about how "they don't make anime like this any more". If that's true, then fuck it.
So... Yeah, there are gamers out there who don't like anime - gimme Western animation with plotlines that are vaguely comprehensible to people without a degree in philosophy and ultraviolence that doesn't involve all the women randomly getting naked any day of the week!