Lets consider this method of finding inverse function:

$$f^{-1}(x) = \sum_{k=0}^\infty A_k(x) \frac{(x-f(x))^k}{k!}$$

where coefficients $A_k(x)$ recursively defined as

$$\begin{cases} A_0(x)=x \\ A_{n+1}(x)=\frac{A_n'(x)}{f'(x)}\end{cases}$$It is evident that for some classes of functions starting from some point $A_k(x)$ becomes zero and thus the inverse function can be expressed in closed form.

For example, the expression has limited number of terms for any function of the following form:

$$f(x)=a \sqrt[n]{x+b}+c$$

where $n\in \mathbb{N}$. It is also evident that there are other classes of functions for which the series have limited number of terms.

So my question is for which other classes of functions this series give inverse function is closed form?