Real life examples of commutative but non-associative operations

Question

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.

Answer 1

Let $\circ$ be the "function" of $a$ and $b$ having a child. Then $$(a\circ b)\circ c \neq a\circ(b\circ c),$$ where I assume asexual reproduction...

Answer 2

Mixing (same amount of ) primary colors:

(red + blue ) + blue = purple + blue = blue purple,

red + ( blue + blue ) = red + blue = purple.

Answer 3

What about commas?

Brian Rushton finds inspiration in cooking, his family and his dog.

vs.

Brian Rushton finds inspiration in cooking his family and his dog.

Shows pretty well how associativity makes a difference.