Well, addition of positive integers is associative and has an obvious real-world analogue, so the associophobic math isn't a good choice for describing reality. But if you lived in Bejeweled, addition wouldn't make much sense as a concept - sometimes pushing a thing close to another thing yields two things, sometimes zero. The most fundamental operation would be "flip a pair of adjacent things", which is not associative. (It's sort of a transposition, which would give you group theory, which is full of associative operations, but I don't think you can factor in disappearing rows while preserving associativity - it destroys the bijectivity.)