I'm.. gonna go with my two favorites, because one is older and as been my favorite since I was 13, the other is one that just aired barely two years ago. So. yeah. XD; I'm gonna do both.
Fushigi Yuugi
1. It's "old"!
2. It's horribly, horribly shoujo.
3. It's nearly a reverse-harem, which plenty of people hate.
4. It's a longer series, with around 70-80 episodes, and plenty of people are turned off by anything that's more than 52 episodes.
5. The dub is really pretty awful, so you'd have to stick with the Japanese audio, unless it was just to bash it.
6. Some people say that the characters aren't built up enough, I suppose that could be true, so I'll go with it, but I myself see nothing wrong with them? They're built up until either the series ends, or they die. >_> (iffy!)
7. "Miaka!" "Tamahome!" The two main love interests say each others names 30 times an episode.
8. The character designs are very unrealistic, and colorful. Fans that prefer more down-to-earth series wouldn't be too pleased.
9. It's corny, and dramatic. Enough said there.
10. "I love you, even if we just met two episodes ago!"
That's what I've heard people say, and what I've realized myself, despite loving this series. XD
Angel Beats
1. This one is super short, could be watched in one sitting, and some people might not have time to even like it very much, let alone mark it as a favorite.
2. Stupid, sarcastic humor are not some people's cup of tea.
3. "WHY IS THIS HAPPENING?" It leaves you wondering that often.
4. God help you if you watch this entire series, and are someone that hates J-pop.
5. "THAT ENDING WAS BULL!" Surprise plot twists, they screw with your mind.
6. "Otonashi is boring.." It's mostly the supporting cast that makes the series.
7. The fans who watch it for the romance end of it have to wait awhile, or maybe just ship un-canon shounen-ai.
8. Identity issues. It's genre seems to go all over the place.
9. "brb shutting myself in my room and crying" This series, humor aside, doesn't have all too many moments of pure happiness, it's mostly bittersweet, and some might find it just depressing.
10. It's animation team is KEY, which sits badly with some anime fans, apparently. -w-
11. The dub is pretty horrible. Really. Really horrible.