Can we construct a successor function from successive applications of those two functions? Let $f(x) = 5 \cdot x + 3$ and $g(x) = \frac{x}{8}$,
Is it possible to construct a function $s$ such that $s(x) = x + 1$ via successive applications of any of f and g?

If I understand correctly, a big composition of $f$ and $g$ involving applying $f$ a total of $k$ times and $g$ a total of $\ell$ times (in any order) will yield a function of the form
$$\frac{5^k}{8^\ell}x + A,$$
where $A$ is a constant that depends on the order of applications. Since $5^k/8^\ell$ can never be equal to $1$, the answer is no.
