In this previous post I asked for the smallest set of continuous real functions that could generate Q by iteration starting from 0. Surprisingly one continuous function suffices.

In the question I gave the example of three rational functions that generate $\mathbb{Q}$, $f(x)=1/x$, $g(x)=x+1$ and $h(x)=x-1$. It would be interesting to know if this is best possible and in particular whether one rational functional can generate all of $\mathbb{Q}$:

Can $\mathbb{Q}$ be generate as the orbit of less than 3 rational functions?

The question Orbits of rational functions asks a more general question but I don't think explicitly answers it for $\mathbb{Q}$ itself.