  For a given positive integer $n$, how long can the process $x \mapsto n\text{
}(\text{mod } x)$ last before reaching $0$? We improve Erd\H{o}s and Shallit's
upper bound of $O(n^{\frac{1}{3}+\varepsilon})$ to
$O(n^{\frac{1}{3}-\frac{2}{177}+\varepsilon})$ for any $\varepsilon > 0$.
