Basically, let the expressibility of an algebra be a term used to describe the power of the algebra's operations to express functions from an algebra to itself.
(Before I move on, I'm going to assume an algebra's operations are closed)
What branch of math is concerned with how expressible functions are with algebraic operations?
For example: the functions of the real numbers to the real numbers are not all expresable with the operations of addition and multiplication (excluding division), for some functions are discontinuous and all polynomials of the reals are continuous.
I am also concerned about functions with an algebra $A$ like $f:A^{n} \rightarrow A$. This includes functions with more than one variable. What's interesting is this concernment also questions if operations can be used to express other operations.
At the most basic level, what you're talking about is a clone: a clone is essentially a family of functions on a give set which is closed under composition" (e.g. if a binary function $f$ and a ternary function $g$ are in my clone, then so must be the five-ary function $(x, y, z, u, v)\mapsto g(f(x, y), f(z, u), f(v, g(x, x, x)))$). A family $\mathcal{F}$ of functions on a set $A$ induces a clone - namely, the smallest clone containing $\mathcal{F}$. One area where clones are studied is universal algebra.
There are more powerful notions of definability, however (e.g. not just using direct composition, but logic - quantifiers, Boolean operations, etc.). At this point model theory comes into play, and there are lots of important theorems of the form "Every function which is definable in this structure has this form." But you seem more interested in the direct-composition approach, so I think universal algebra is the place to look.