Aqueous suspension of nanoclay Laponite undergoes structural evolution as a function of time, which enhances its elasticity and relaxation time. In this work, we employ an effective time approach to investigate long-term relaxation dynamics by carrying out creep experiments. Typically, we observe that the monotonic evolution of elastic modulus shifts to lower aging times, while maxima in viscous moduli get progressively broader for experiments carried out on a later date after preparation (idle time) of the nanoclay suspension. Application of effective time theory produces a superposition of all the creep curves irrespective of their initial state. The resulting dependence of the relaxation time on aging time shows very strong hyper-aging dynamics at short idle times, which progressively weakens to demonstrate a linear dependence in the limit of very long idle times. Remarkably, this behavior of nanoclay suspensions is akin to that observed for polymeric glasses. Consideration of aging as a first-order process suggests that continued hyper-aging dynamics causes cessation of aging. The dependence of relaxation time on aging time, therefore, must attenuate eventually producing linear or weaker dependence on time in order to approach a progressively low-energy state in the limit of very long times as observed experimentally. We also develop a simple scaling model based on a concept of aging of an energy well, which qualitatively captures various experimental observations very well, leading to profound insight into the hyper-aging dynamics of nanoclay suspensions.