We study the interaction of intense petawatt laser pulses with overdense plasma over several picoseconds, using two- and three-dimensional kinetic particle simulations. Sustained irradiation with non-diffraction-limited pulses at relativistic intensities yields conditions that differ qualitatively from what is experimentally available today. Nonlinear saturation of laser-driven density perturbations at the target surface causes recurrent emissions of plasma, which stabilize the surface and keep absorption continuously high. This dynamics leads to the acceleration of three distinct groups of electrons up to energies many times the laser ponderomotive potential. We discuss their energy distribution for applications like the fast-ignition approach to inertial confinement fusion.