We show that for a given base $b$ and a proper subset $E \subset \{0, \dots, b-1\}$, #$E < b-1$, the set of numbers $x \in [0,1]$ which have no digits from $E$ in their expansion to base $b$ consists almost exclusively of $S^*$-numbers of type $\leq 2$. We also give upper bounds on the Hausdorff dimension of some exceptional sets.