Let $A$ be the smallest set of the following functions on the positive reals:

- The identity function is in $A$,
- for a function $f\in A$ also the inverse $f^{-1}$ is in $A$,
- for two functions $f,g\in A$ also the sum $f+g$, the product $f\cdot g$ and the composition $f\circ g$ is in $A$.

In each generation the following properties are conserved: each function is strictly increasing and maps the positive reals surjectively to the positive reals. That's why we always can take the inverse above. The set $A$ is a group with respect to the composition operation.

The set $A$ contains all the polynomials with positive integer coefficients and zero constant term. Each non-trivial polynomial has finitely many fixpoints. Having finitely many fixpoints also implies being linearly orderable by "orders of infinity" (Hardy), i.e. by $f<_\infty g$ if there is an $x_0$ such that $f(x)\lt g(x)$ for all $x\gt x_0$ .

Do all the functions of $A$, except the identity function, have finitely many fixpoints?