Just How Strong is Associativity?

A particularly notorious example of non-associative binary operations are addition and multiplication of numbers - when done in a computer using floating point arithmetic. That is, the results of x = a+(b+c); and x=(a+b)+c; may be very different. One needs to be careful in programming additions to avoid excessive rounding errors in such situations.

For all associative binary operations on a set, there is a faithful representation as a sub-semigroup of the semigroup $(X^X,\circ)$ for some set $X$.

That is the nature of associativity, on some level - it is function composition.

This puts us into the area of category theory, too. It would be probably not very useful to do category theory without associativity of composition.

The most basic place I've seen non-associativity is in $\lambda$-calculus and/or combinatory logic, where $ab$ represents application of $a$ to $b$.

If you look at lambda calculus, you can think of it as $a\star b=\phi_a(b)$. That's actually true for all binary operations, of course - we can define $\phi_a(b)=a\star b$. For associative operators, however, we have the lovely feature $\phi_a\circ \phi_b = \phi_{a\star b}$.