  Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We prove
that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly
greater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple',
in a suitable sense. Our result applies, among other classical numbers, to
badly approximable numbers, non-zero rational powers of ${\mathrm e}$, and
$\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large.
It establishes an unexpected connection between the irrationality exponent of a
real number and its $b$-ary expansion.
