Rayleigh-Taylor instability growth is shown to be hydrodynamically scale invariant in convergent cylindrical implosions for targets that varied in radial dimension and implosion timescale by a factor of 3. The targets were driven directly by laser irradiation providing a short impulse, and instability growth at an embedded aluminum interface occurs as it converges radially inward by a factor of 2.25 and decelerates on a central foam core. Late-time growth factors of 14 are observed for a single-mode m=20 azimuthal perturbation at both scales, despite the differences in laser drive conditions between the experimental facilities, consistent with predictions from radiation-hydrodynamics simulations. This platform enables detailed investigations into the limits of hydrodynamic scaling in high-energy-density systems.