Top of my list for this is Mickey Smith, Rose's selfish, unambitious, control freaky boyfriend. He does grow, admittedly, as the series develops, but he never gets to the point where I can say that he doesn't irritate me. He could cope with Rose going off with Nine but as soon as he realised, in Boom Town, that Captain Jack was in the TARDIS, he felt threatened. So we can add superficiality and shallowness to his qualities.
Speaking of Captain Jack, yes, I love him, and so does everyone else, but even his best friends would consider him annoying at times, wouldn't they? Smugger than smug, fickle, voraciously omnisexual, vain, but very funny, charming and unconventional.
Mel. Screaming. Need I say more?