Faà di Bruno's formula is a formalism that generalizes the chain rule of two functions to higher derivatives. I suspect that it is not often taught to undergrads because one can iteratively apply the simpler rules to the same effect, but over time I've find the more complicated generalization useful. While Faà di Bruno's formula gives us an expansion for $\frac{d^n}{dx^n} f(g(x))$, is there an expansion for the more general case of finding $$\frac{d^n}{dx^n} [f_1 \circ f_2 \circ \cdots f_{m-1} \circ f_{m}]$$
where
$$\{f_1, f_2, \cdots f_{m-1}, f_{m}\}$$
is a finite collection of $m$ functions?
Footnote: Assume that these functions have sufficient smoothness.