The study of so-called exceptional sets goes back to the work of Jarnik-Besicovitch who showed that the Hausdorff dimension of the set of badly approximable numbers on the real line is 1. Seen in a more general context, given a continuous map of a compact metric space and some subset $A$, an $A$-exceptional set is defined to be the set of points whose orbit does not accumulate at $A$. We study the "size" of such sets in terms of their Hausdorff dimension and topological entropy. Here we consider a quite general context of $C^{1+\varepsilon}$ maps with sufficient hyperbolicity. If the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension and the topological entropy of the $A$-exceptional set both are large. This is joint work with S. Campos.