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:

$$
(1-2)-3 = -4
$$
$$
1-(2-3) = 2
$$
IEEE-754 floating point, due to the rounding inherent in operations.
a + b involves a rounding step.
It's one of the reasons fused multiply-adds exist.
0.7 + 0.1 + 0.2 from here:
https://stackoverflow.com/questions/10371857/is-floating-point-addition-and-multiplication-associative