In my job of helping with academic papers I came across a nice example of nonassociativity. It's the nonassociativity of adjectives.
There is a demographic breakdown by race in terms of non-Hispanic-White, non-Hispanic-Black, non-Hispanic-Asian, Hispanic, etc. I need to compare non-Hispanic-White with everybody else. So then the term would be "non-non-Hispanic-White" (!) I probably can't use that but it made me realize that we would need parentheses because (non (non-Hispanic (White))) means something different than ((non (non))(Hispanic(White)) and so on. And, in general, if you string together adjectives or adverbs this issue can come up.
I wonder what adjectives and adverbs correspond to in math... But clearly Negation is one. And perhaps also quantifiers, Some, All... and other parts of logic? Conjunctions?