The Rayleigh-Taylor unstable growth of laser-seeded, 3D broadband perturbations was experimentally measured in the laser-accelerated, planar plastic foils. The first experimental observation showing the self-similar behavior of the bubble size and amplitude distributions under ablative conditions is presented. In the nonlinear regime, the modulation sigma(rms) grows as alpha(sigma)gt(2), where g is the foil acceleration, t is the time, and alpha(sigma) is constant. The number of bubbles evolves as N(t) alpha(omegat sq.rt(9) + C)(-4) and the average size evolves as <lambda> (t) alpha omega(2)gt(2), where C is a constant and omega = 0.83 +/- 0.1 is the measured scaled bubble-merging rate.