Colloidal dispersions consist of small particles that are immersed in a molecular fluid. The particles move by diffusion, driven by the thermal motion of the molecules surrounding them. However, as the particle concentration increases, the diffusion of the particles becomes increasingly hindered due the presence of their neighbors; consequently, the structural relaxation time, describing the time-scale over which the system reconfigures, increases. This structural relaxation time appears to diverge at a critical concentration that defines the colloidal glass transition, where solid-like behavior sets in. We use new dynamic light scattering methods to probe structural relaxation processes of systems composed of deformable spheres extending the range of volume fraction investigated to deeply quenched systems, far beyond the glass transition. Our investigations reveal that structural relaxation processes are never completely suppressed. Indeed, our results suggest that imbalanced internal stresses within the system become significant contributors to the dynamics of very concentrated dispersions, such that structural relaxation persists beyond the colloidal glass transition even though thermal effects are no longer relevant.