In Gen 1, Brock in Yellow because it's such a hassle to either try winning with Pikachu, train a Caterpie to Lv. 10 and spam Confusion until you win, or catch a Mankey, which I never realized I could do.
Misty's Starmie is really annoying, but she's overall not that bad, unlike a certain gym leader I haven't mentioned yet that I think we all know about.
Sabrina and Psychic types that pretty much have no weakness in Gen 1, they're really annoying.
Clair has that annoying Kingdra, but it's not as bad as Juan's in Emerald. We all know the one. Even after that, she's such a sore loser and refuses to give you a badge, but in Crystal, if you play your cards right, you can get a free Extremespeed Dratini.
Then, there's her. Whitney, the scourge of just about every Gen 2 player's childhood. Well, not Wintney as a person, but her evil cow of death, Miltank. I don't think I need to say any more since all the other posts speak for me about that demon cow.