We report on near-bottom waves of elevation with amplitude nearly half the 25 m water column, very steep, and propagating into a sheared turbulent wave-guide. The waves contained trapped cores transporting parcels of water shoreward. These large waves depart strongly from weakly-nonlinear and weakly-nonhydrostatic conditions and challenge established paradigms. They can also represent an important factor in the across-shore transport of plankton and contaminants.