On the expansion of some exponential periods in an integer base

We derive a lower bound for the subword complexity of the base-$b$ expansion ($b\ge 2$) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to Ferenczi and Mauduit. As a consequence, we obtain the first lower bound for the subword complexity of the number $e$ and of some other transcendental exponential periods.