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.

Let $$∘\circ$$ be the “function” of $$aa$$ and $$bb$$ having a child. Then
$$(a∘b)∘c≠a∘(b∘c),(a\circ b)\circ c \neq a\circ(b\circ c),$$