We study periodically driven bosonic scalar field theories in the infinite limit. It is well known that the free theory can undergo parametric resonance under monochromatic modulation of the mass term and thereby absorb energy indefinitely. Interactions in the infinite limit terminate this increase for any choice of the UV cutoff and driving frequency. The steady state has nontrivial correlations and is synchronized with the drive. The model at infinite provides the first example of a clean interacting quantum system that does not heat to infinite temperature at any drive frequency.