What generally turns me off to anime is that a lot of the plots can be boiled down to: a quest for revenge OR to prove I'm the strongest. Now I must stress that this is not all of them, but certainly enough to make up a significant portion of the base.
Furthermore, what bothers me the most is a lack of relatable characters. I was watching this one series -Claymore- which I thought started out pretty good. Here's this chick who can barely keep up with this monstrosity and is basically getting her ass kicked until someone distracts the thing and she kills it, but by the end of the series she's so frickin' over-powered that I'm not exactly sure what's a threat to her or not. She's basically operating on the level of a minor Greek god, and I've lost total investment in her character. Same issue with Van Helsing, and Dragon Ball.
I also gave another series 'gunslinger girls' a shot, but had to quit because I encountered the same problem in a different context. The characters basically acted like unfeeling robots, and I just can't bring myself to care about a plot that boils down to 'they needed to do this thing, so they did.'(over-arching plot of the series not withstanding because I wasn't able to make it past Episode 3)
Alternately, I LOVED Desert Punk, I LOVED FMA, I LOVED The Last Airbender (Yes, I know that's Amerime now ask me if I care). These were all about the characters; learning, growing, fucking up, making difficult decisions, and even though Aang and Edward basically had superpowers it was still pretty clear what the limits of their respective abilities were in any given context.
Give me more anime like that!