We show that anomalous Floquet topological insulators generate intrinsic, non-Hermitian topology on their boundaries. As a consequence, removing a boundary hopping from the time-evolution operator stops the propagation of chiral edge modes, leading to a non-Hermitian skin effect. This does not occur in Floquet Chern insulators, however, in which boundary modes continue propagating. The non-Hermitian skin effect on the boundary is a consequence of the nontrivial topology of the bulk Floquet operator, which we show by designing a real-space topological invariant. In this paper, we introduce a form of mixed higher-order topology, providing a bridge between research on periodically driven systems and the study of non-Hermiticity. This suggests that periodic driving, which has already been demonstrated in a wide range of experiments, may be used to generate non-Hermitian skin effects.