Hmm. A lot of different examples have already been posted so I'll go for something more broad that is probably going to be pretty contentious. As much s I hate using the phrase "anime style" I can't help it when the majority of popular anime
do have a style. Not that the particular way of character animation is bad, per say, but it is overused to all hell and is actually dampening what anime as a whole can achieve.
Now what is "anime style"? Well to put it simply, it is the way characters are drawn in 90+% of popular anime. To give an example,
http://fc05.deviantart.net/fs27/f/2008/142/5/e/Conglomerate_Of_Anime_shows_by_SoulOfSolaria.jpg.
Now, I can not stress enough that not all anime fits this style. To give a recent example, Princess Kaguya was incredibly popular among anime fans, and for the most part broke this style. Furthermore, there are an insane amount of anime that are less popular that do not follow the style either. It's why I hate calling anime a "style". Unfortunately, many of the remaining anime do in fact prefer to stick to what I like to call style tropes. People are drawn in a certain way, using a certain form of animation because it is what is popular. The style then affects the way the anime looks, the way dramatic and action scenes play out, the way the characters act, and a whole slew of other factors.
It has been stated that anime is the most versatile of all the different forms of animation. I'd agree with that sentiment, but I can't help it when I see the same style/tropes come on that I wan't them to diversify. It sucks that if I want something that looks and acts different in Japan I have to go into the underground scene. For anyone saying that it works like that in any medium, sure. That doesn't mean I shouldn't worry about trying to help a part of this one grow.