An iguana writes the number 1 on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\frac{1}{x}$ or $x+1$. Can the iguana eventually write the number $\frac{20}{17}$ ?
Yes. First, the iguana writes

$$
1 \rightarrow 2 \rightarrow \frac{1}{2} \rightarrow \frac{3}{2} \rightarrow \frac{2}{3}
$$

Then, the iguana adds 1 to arrive at $\frac{17}{3}$. Finally, finish with

$$
\frac{17}{3} \rightarrow \frac{3}{17} \rightarrow \frac{20}{17} .
$$

In fact, see if you can prove that any positive rational number can be achieved!