There is one kind of character I hate, and that's the "No matter what, I'll pull through, with the help of friendship, and I'll make the world better for everyone!" bullshit character that doesn't re-evaluate their world view, ever, and is so insufferably optimistic that even when they do that thing where they've just suffered a terrible loss and the bad guy is defeated in revenge, and the bad guy's sympathetic motive is explained through a cynical and self-sacrificing worldview and the protagonist is confronted with it, they give a speech that reiterates their naive optimism and they crack a sad smile that's meant to make you see what a truly honest and caring person they are, I still fucking hate them. Especially when they have a backstory where they undergo hardship and are ostracised from the rest of society. I'm looking at you, Naruto. You piece of s-
Ichigo Kurosaki fulfills the same narrative role (sticking to his guns, ramping up inner strength and working tirelessly to destroy the bad guys), but is a lot more down-to-business about it and never goes on about happiness or any of that fairytale crap. If the manga is anything to go by at least.
Another thing which annoys me, but doesn't necessarily prevent me from liking the character, but will break immersion a little, is the character that can predict the future or instantly analyse a person because "they're just that smart". Bullshit. No-one can do that. Characters that fall into this category include Lelouch (although he's still a favourite of mine for other reasons) and Kougami's psychologist mentor from Psycho-Pass (who is ridiculous in the instant analysis department). Contrast Sumeragi Lee Noriega, in combination with not being ridiculously sure of herself and occasionally losing, it creates a feeling that she's not omnipotent, but every strategist in the series is at the edge of their abilities rather than coasting along.
EDIT: And since I've just finished watching Psycho-Pass again, good character: Akane Tsunemori. Naive and idealistic at first, but always understandable, and ends up retaining her ideals but has a much different and more serious worldview, and even her one seemingly arbitrarily naive goal is grounded and understandable (not spoiling). This is what I like to see. Actual character development through hardship, as opposed to the relative cop-out of saying "they're just the most forgiving person in the world", which now that I look at it, is basically just an excuse to never make any real challenging ideological point.