Not annoying per se, but just creepy. And not the cool kind of creepy either; the unsettling kind of creepy where you feel the impulse to stay at least 4m away from them at all times.
A few months ago this guy moved in to the apartment next to the one me and my mom live. We don't really interact too much, but one of the only times he did, he had a sort of off-putting feel to him and he kept giving sort of backhanded compliments. He also stares directly into the apartment (the front door is made of glass) whenever he passes by, which is for obvious reasons weird.
For a more properly "annoying" neighbour, back around the time I joined TBT and was living in New York, this we had this weird old lady who lived above us. She always shouted at us from her apartment when we were leaving for school. Much as I appreciate the sentiment, Sandra, it gets irritating fast. She also had this weird thing where she just... threw bread on the roof to feed the wildlife? Needless to say we wound up getting a few problems with roaches and rats when we lived in that complex, and I think we can all guess why that was. Another weird thing about her was that she was kind of racist too; she seemingly used our neighbours as a scapegoat for things like the parking lot being dirty (for reference, there was a Latino family living in the apartment building next to ours, and both buildings shared a parking lot), which just kinda sits wrong with me.