Let $u_0 \ge 2$ be a rational, and $u_{n+1}=⌊u_n⌋(u_n - ⌊u_n⌋ + 1)$.
Question: Does the sequence $(u_n)$ reach an integer?
It was checked by computer (with SageMath) for $u_0=\frac{p}{q}$ with $p \le 32000$.
Example: if $u_0=\frac{11}{5}$ then $(u_n)= (\frac{11}{5}, \frac{12}{5}, \frac{14}{5}, \frac{18}{5}, \frac{24}{5}, \frac{36}{5}, \frac{42}{5}, \frac{56}{5}, \frac{66}{5}, \frac{78}{5}, 24, \dots)$.
A positive answer would provide an alternative to continued fraction theory (see cross-post here).