In this Letter, we show through numerical simulations and analytical results that overlapping multiple (N) laser beams in plasmas can lead to strong stochastic ion heating from many (~N(2)) electrostatic perturbations driven by beat waves between pairs of laser beams. For conditions typical of inertial-confinement-fusion experiment conditions, hundreds of such beat waves are driven in mm(3)-scale plasmas, leading to ion heating rates of several keV/ns. This mechanism saturates cross-beam energy transfer, with a reduction of linear gains by a factor ~4-5 and can strongly modify the overall hydrodynamics evolution of such laser-plasma systems.