In this paper, we develop an analytical framework for the statistical analysis of the battery recharging time (BRT) in reconfigurable intelligent surfaces (RISs)-aided wireless power transfer (WPT) systems. Specifically, we derive novel closed-form expressions for the probability density function (PDF), cumulative distribution function, and moments of the BRT of the radio frequency energy harvesting wireless nodes. Moreover, a closed-form expression of the PDF of the BRT is obtained for the special case when the RIS consists of a large number of elements. Capitalizing on the derived expressions, we offer a comprehensive treatment for the statistical characterization of the BRT and study the impact of the system and battery parameters on its performance. Our results reveal that the proposed statistical models are analytically tractable, accurate, and efficient in assessing the sustainability of RIS-assisted WPT networks and in providing key design insights for large-scale future wireless applications. For example, we demonstrate that a 4-fold reduction in the mean time of the BRT can be achieved by doubling the number of RIS elements. Monte Carlo simulation results corroborate the accuracy of the proposed theoretical framework.