Non-associative operations

There are lots of operations that are not commutative.

I'm looking for striking counter-examples of operations that are not associative.

Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of genuinely lesser importance?

Which role do algebraic structures with non-associative operations play?

There's a big gap between commutative and non-commuative algebraic structures (e.g. Abelian vs. non-Abelian groups or categories). Both kinds of algebraic structures are of equal importance. Does the same hold for assosiative vs. non-associative algebraic structures?

Subtraction:

$$ (1-2)-3 = -4 $$ $$ 1-(2-3) = 2 $$

A simple example, and one that even elementary school students should be able to understand, is averaging.

average(average(a,b),c)

and

average(a,average(b,c))

are, generally, not equal to each other.