We experimentally study the late time, highly nonlinear regime of the Rayleigh-Taylor Instability in a decelerating phase. A series of laser-driven experiments is performed on the LULI2000 laser, in which the initial Atwood number is varied by adjusting the decelerating medium density. The high power laser is used in a direct drive configuration to put into motion a solid target. Its rear side, which initially possesses a two-dimensional machined sinusoidal perturbations, expands and decelerates into a foam leading to a Rayleigh-Taylor unstable situation. The interface position and morphology are measured by time-resolved x-ray radiography. We develop a simple Atwood dependent model describing the motion of the decelerating interface, from which its acceleration history is obtained. The measured amplitude of the instability, or mixing zone width, is then compared with late time acceleration dependent Rayleigh-Taylor instability models. The shortcoming of this classical models, when applied to high energy density conditions, is shown. This calls into question their uses for systems, where a shock wave is present, such as those found in laboratory astrophysics or in Inertial Confinement Fusion.