The nonlinear saturation amplitudes attained by Rayleigh–Taylor perturbations growing on ablatively stabilized laser fusion targets are crucial in determining the survival time of those targets. For a given set of baseline simulation parameters, the peak amplitude is found to be a progressive function of cross‐sectional perturbation shape as well as of wave number, with three‐dimensional (3‐D) square modes and two‐dimensional (2‐D) axisymmetric bubbles saturating later, and at higher amplitudes than two‐dimensional planar modes. In late nonlinear times hydrodynamic evolution diverges; the 3‐D square mode bubble continues to widen, while the 2‐D axisymmetric bubble fills in.