I asked this on MathStackExchange but didn't get an answer, so I'm trying it here:

Let $\alpha$ be an algebraic number and denote with $\frac{p_n}{q_n}$ the $n$-th convergent of $\alpha$ that we get when expressing $\alpha$ as continued fraction. Is it true that $\underset{n \rightarrow \infty}{\lim} \frac{1}{n} \log(q_n)$ is algebraic?