On the length of Pierce expansions

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