Real life examples of commutative but non-associative operations I've been trying to find ways to explain to people why associativity is important.
Subtraction is a good example of something that isn't associative,  but it is not commutative.
So the best I could come up with is paper-rock-scissors; the operation takes two inputs and puts out the winner (assuming they are different).
So (paper rock) scissors= paper scissors = scissors,
But paper (rock scissors)= paper rock = paper.
This is a good example because it shows that associativity matters even outside of math.
What other real-life examples are there of commutative but non-associative operations? Preferably those with as little necessary math background as possible.

The averaging operation, defined by $$a\oplus b= \frac{a+b}2$$ is commutative but not associative.
