Two examples of operations are infinite conjunction and disjunction on set of propositions: to be correct this should be regarded more like a family of operations.
Let $\{0,1\}$ be the set truth values, for each cardinal $\aleph$ you can consider the operation $\bigwedge \colon \mathcal \{0,1\}^\aleph \to \mathcal \{0,1\}$ this operation is such that for every family of truth values $(x_i)_{i \in \aleph} \in \{0,1\}^\aleph$, $\bigwedge_{i \in \aleph} x_i = 1$ if and only if $x_i=1$ for each $i \in \aleph$.
In similar way you can consider the operation $\bigvee \colon \{0,1\}^\aleph \to \{0,1\}$ such that for every family $(x_i)_{i \in \aleph} \in \{0,1\}^\aleph$ the equality $\bigvee_{i \in \aleph} x_i = 1$ if $x_i=1$ for at least one $i \in \aleph$.
Another operation which is important (at least in my opinion) is the juxtaposition of words. Consider an arbitrary set $\Sigma$ then you can define over the set $\Sigma^*=\bigcup_{n \in \mathbb N} \Sigma^n$ the operation $\cdot\colon {\Sigma^*}^2 \to \Sigma$ defined as $\cdot \left((x_i)_{i=1,\dots,n},(y_i)_{i=1,\dots,m}\right) = (z_i)_{i=1,\dots,n+m}$ where $z_i = x_i$ if $i \leq n$ and $z_i = y_{i-n}$ otherwise. As I said this is a pretty important operation because the set $\Sigma^*$ with this operation gives us an example of free monoid, which is in a certain sense a prototypical monoid, in which we are able to explicit write computations.