  Non-Hermitian systems exhibit nontrivial band topology and a strong
sensitivity of the energy spectrum on the boundary conditions. Remarkably, a
macroscopic number of bulk states get squeezed toward the lattice edges under
open boundary conditions, an effect dubbed the non-Hermitian skin effect
(NHSE). A well-established dynamical signature of the NHSE in real space is the
directional bulk flow (or persistent current) for arbitrary initial excitation
of the system, which is observed at long times. Here we unravel a different
dynamical signature of the NHSE in real space that manifests itself in the {\em
early-time} dynamics of the system, namely self-acceleration of the wave
function. Self-acceleration is demonstrated to occur rather generally in
single--band lattice models probed by single-site excitation, where the
acceleration turns out to be proportional to the area enclosed by the energy
spectrum of the Bloch Hamiltonian under periodic boundary conditions. The
observation of wave packet self-acceleration at early times is a clear
signature of the NHSE and should be experimentally accessible using synthetic
non-Hermitian matter, for example in discrete-time photonic quantum walks.
