What does this mean:
This slowly becomes annoying at best
I cannot understand what the last two words of this phrase signify.
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It means that the maximum positive quality of the subject ("this") is to be annoying; that is the most it can ever achieve. The likelihood is that it is worse than annoying, since the odds of a thing being at its best all the time are small.
Often constructions like this are counterbalanced with a "worst case" scenario. For example,
This is annoying at best, and at worst it's downright destructive.
"At best" means that whatever has just been mentioned is considered to be the least detrimental of all probable ramifications or outcomes of a particular event/scenario. Appending the words at best to the end of a sentence implies that you expect much more serious or damaging things to occur as a result of something.
"At best" used in this sentence means that, the nicest way to describe the thing is "annoying".
It's that bad, that the most favourable thing about it is, it's annoying.
This can be compared with "at worst", when describing situations e.g.
At worst, we could only die.
This is stating that the worst possible outcome, is to die. In the same way, "at best" is describing that the most desirable outcome, or most positive description, of something or someone, is "annoying."