In essence, I am looking for an example of a semigroup or a semicategory (closure is not that important, but it is useful) that is NOT a monoid or category.
Hopefully, there is a neat and simple-to-understand example of an only-associative operation on an infinite set.
Edit: Also, non-commutative!
Edit #2: From all the answers, it seems there are no associative, non-identity, non-inverse operations on the reals such that for $x,y \in \mathbb{R}$ the operation is $x \cdot y = f(x,y)$ where $f(x,y)$ is a some analytically expressible function, such that the reals form a semi-group under that operation.
Best Answer
How about strictly upper triangular matrices with matrix multiplication? Matrix multiplication is clearly associative and the product of two strictly upper triangular matrices is again strictly upper triangular. There are no inverses since these matrices have nontrivial kernel and there is no identity since the identity matrix is not strictly upper triangular.