Is there a word for an annoying person? I'm not just looking for synonyms of annoying.
It would depend on how they're annoying.
If they are annoying in a little sister or little brother sort of way, try "pest".
If they are annoying in an unwanted suitor way, especially if 'suitor' is very much a euphemism, try "creep".
If they are annoying in a general way, they may be a "loudmouth" (self-evident), a "nerd" (annoying in that slow-to-develop socially way), or even a "busybody" (in this case, more in the personally nosy way than in the gossipy way).
In general, all of these people are "nuisances".
(But these are also given as nouns, and most won't work as direct substitutes for the adjective annoying.)
irksome TFD adj.
Causing annoyance, weariness, or vexation: irksome duties; irksome restrictions;
an irksome person