  We consider the real number $\sigma$ with continued fraction expansion $[a_0,
a_1, a_2,\ldots] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots]$, where $a_i$ is
the largest power of $2$ dividing $i+1$. We compute the irrationality measure
of $\sigma^2$ and demonstrate that $\sigma^2$ (and $\sigma$) are both
transcendental numbers. We also show that certain partial quotients of
$\sigma^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and
Wilson.
