Personally, I love the games-- Pokemon to me is one of the best series of video games, but some things do drive me crazy.
From the game:
1. How you needed to have multiple versions of the same game to get different Pokemon. (BLUE & RED / GOLD & SILVER & PLATINUM, etc...) Even more annoying, in Diamond and Pearl, you actually needed to plug in your old games to get other Pokemon. Why is Bulbasaur not in the wild!? Why can't I fricken catch Pidgey in Diamond in some field!?!?!
2. The one reason that they use to explain why you can't catch a Pidgey in Diamond is because they are in a different region in the world. How about being able to travel to different fricken regions, dammit!! In Gold/Silver, you could travel between Johto and Kanto. They should have continued it like this for the 3rd and 4th generation.
3. With all the high-tech gadgetry, you'd think they'd have the fricken internet where Ash could download all the Pokemon data from around the world, instead of being surprised when he sees a Taillow.
From the show:
1. Ash saw some of the most amazing Pokemon and never caught them. (some of you guys already mentioned this, but you're absolutely right).
2. Team Rocket's obsession with Pikachu. Come on. You'd think they'd give up after their consecutive 500th failure. Have some self-respect! Damn, they were so annoying after the first season...