Basic examples of monoids? What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they are groups...
I've read this and this and some of this, but would like more examples that presuppose little to no algebra.

A generic answer is the monoid of all functions from a set $E$ into itself under the composition of functions. This example is generic since every monoid is isomorphic to a submonoid of such a monoid. In particular, take any set of functions from $E$ to $E$ and close under composition: you will get a monoid. See this link. This type of example occurs frequently in automata theory.
You could also consider partial functions or relations on $E$, still under composition. The monoid of $n \times n$ matrices over a ring under the usual multiplication of matrices is also a quite natural example. If you have a monoid $M$, the set $\mathcal{P}(M)$ of all subsets of $M$ is also a monoid under multiplication defined by
$XY = \{ xy \mid x \in X, y \in Y \}$ (where $X, Y \in \mathcal{P}(M)$).
