It ultimately all comes down to this: Which character in the show is NOT annoying?
And that, my friends, is a very hard question. One which I shall attempt to answer. During my long and hard research, I have concluded that, to this day, there have been only 2 characters or Pokémon which have been very entertaining and not at all annoying, these are:
Mew, from the 1st movie.
Mirage Mew, from the special.
Sadly, the Mew from the 8th movie is not in this list, because it transformed into Pikachu once, BAD MOVE.
That is all.
There are, besides these 2, some Pokémon which were fun, cute, cool and or barely annoying, like Koga's Venomoth (or any Venomoth for that matter), any Wigglytuff, any Flareon (although his trainer during his 1st appearance might as well drop dead), that Hypno from the episode with the Drowsee, and some others. But then again, the Pokémon don't tend to be the most annoying aspect of the show, au contraire. And since this thread refers to characters, I will now try to find the least annoying character.
We have a few viable candidates:
Giovanni, for being the only person in the world of Pokémon with some sense of perspective, someone who knows what he's doing and who's actually doing it right, only hampered by the failure of others. He has also failed to ever loose to Ash, was able to control one of the mightiest Pokémon (you know, the wannabe mew), and has also not yet said anything even remotely silly (feel free to correct me on this one). He also has a Cloyster, so that's like a massive plus.
The downside here, whilst minor, is that Team Rocket is totally abusing him with their silly fantasies.
Will, that totally awesome Psychic-Pokémon training Jotho E4 dude with the awesome costume, for not appearing in the anime at all! Hooray!
Gary, for having a personal group of cheerleaders and for actually knowing what he's doing. Him becoming friends with Ash, on the other hand, is a major downside.
and that is all.
But on the subject of most annoying character: Ash, for appearing in most of the episodes (like, all of them), and yet he has never managed to entertain me even the slightest bit. The lack of a functional brain also isn't really helping his case.
Seriously, even when only his legs are show in those Pikachu Specials he's annoying...
And now you know.