Suppose we define a variadic function in terms of a binary operator $*$ (where the operator $*$ is not necessarily associative and not necessarily commutative): $$f(k, x_1, x_2, x_3, ..., x_k) = (((x_1 * x_2) * x_3) *\ ...)*x_k,$$ $$1 < k$$ So that $f$ applies the same operation to successive partial results. How does one describe the interpretation of $f\ $'s arguments by $f\ $? I want to say "$f$ is ___" or "$f$ has the property ___." What goes in the blank (___)?
For certain $*$ (e.g., the addition operator, $+$), the concept is related to commutativity. But I am looking for something more general, that has nothing to do with the result of $f$ (e.g., for different permutations of arguments), and that would, for example, also apply if $*$ were the subtraction operator ($-$).
This idea is closely related to Common Lisp's reduce, Python's functools.reduce, and R's Reduce.
I don't think there's a snappy term for this. That said, let me try to describe the situation a bit more clearly.
It sounds like you're looking for a term to describe those functions built by (a certain kind of) self-composition. I'll take a stab at rephrasing your question a bit more precisely as follows:
I don't know of a term for this, but "compositional power" wouldn't be too bad ... except that it's somewhat ambiguous. Let me explain why:
For unary functions things are pretty easy: "$n$-fold composition of $f$ with itself" can really only mean one thing. So the analogy with powers in the usual sense is uncomplicated (and indeed we can re-use some language).
However, once the arity goes up we have multiple kinds of self-composition. For example, should "subtraction composed with itself" look like $$(a-b)-c$$ or $$a-(b-c)?$$ In the way I've rephrased your question above I've implicitly assumed that we want to associate to the right, but why shouldn't we want to associate to the left instead, or "the middle" somehow?
A more subtle dissimilarity crops up when we think about the types of objects involved. When we compose two unary functions we get another unary function, but here we're composing in such a way that the arity goes up ($g(x_1,g(x_2,x_3))$ is ternary). So it all just seems messier, and "power" stops being a very convincing term.
(And of course while you've specifically asked about binary functions in your question, there's no reason we can't consider $f$s which are built by self-composing higher-arity functions, and in general there are $(n-1)$ ways to plug an $n$-ary function into itself.)
That said, there is a relevant term lurking here: clone. A clone is a collection of functions - of arbitrary arities - on some set $X$ which is closed with respect to all possible forms of composition and also contains all the projection functions (maps of the form $(x_1,...,x_n)\mapsto x_k$). Given a function $f$ on a set $X$, we can form the clone generated by $f$, and this contains all possible "compositional powers" of $f$. Note that this doesn't exactly correspond to what you want, since it allows "variable reusage:" e.g. if $f$ is a binary function, then the clone generated by $f$ will include the ternary function $$(x,y,z)\mapsto f(f(f(z,x), f(f(x,y), f(y,x))), f(z,z)).$$ This is rather different from the "straightforward" compositions you seem to be looking at, so it's worth pointing out explicitly.