1. Over-used character archetypes. When your characters can be summed up in one word, you've done something wrong.
Moe.
These kinda characters piss me off to no end. Round-faced, air-headed, 'adorable' catchphrases like: 'nyu', 'ugu' or the accursed 'desu'. Irritating and lifeless personalities, standing only as two-dimensional characters for fanservice or eyecandy. Or, for the most part: no reason whatsoever. Though Moe characters with Yandere personalities makes for much fun time [http://media.photobucket.com/image/yandere/stifler-leonheart/yandere.jpg].
Tsunderes.
While not as annoying as moe characters they can still be tiring to watch. Constantly in denial of her feelings and acting tough in front of her crush, tongue-lashings, apathy and bruises await any guy wanting to get close to her. Eventually she lets her guard down and becomes 'dere-dere' or misty-eyed and soft. Basically a pushover when it comes to love. When done well these sort of characters CAN be entertaining to watch, but they're so overused you're begging for some originality.
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2. The 'Will they - won't they?' Storyline.
Guy has hots or girl.
Girl has hots for guy.
Do they do anything about it for the majority of the story?
No.
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3. Memories, or rather the lack of them.
You remember stuff when you were 10 or even younger right? All the turning points in your life, surprises and disappointments. Not for many animé characters, nothing. Back-story drives some characters while some don't seem to have any at all.
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4. The lack of parents.
'Yep, I'm your average fifteen year old child living on his own/with friends. Nothing wrong about that' We usually get the half-assed explanation that they're working over-seas or died in a car crash or MURDERED. (ZOMG DRAMA!!1) It's a couple more characters the animators would have to draw so having them out of the picture makes things easier for them.
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Hotsprings filler, 'There's the bad guy! Get him! *defeats bad guy* There's another bad guy!' Repeat ad nauseam. The RPG-esque way the bad guys get stronger with each arc, forcing the good guys into training, e.g, grinding. And pretty much everything else said above me.
I love animé, but there are a few things which irk me sometimes.