Understanding the charging kinetics of electric double layers is of fundamental importance for the design and development of novel electrochemical devices such as supercapacitors and field-effect transistors. In this work, we study the dynamic behavior of room-temperature ionic liquids using a classical time-dependent density functional theory that accounts for the molecular excluded volume effects, the electrostatic correlations, and the dispersion forces. While the conventional models predict a monotonic increase of the surface charge with time upon application of an electrode voltage, our results show that dispersion between ions results in a non-monotonic increase of the surface charge with the duration of charging. Furthermore, we investigate the effects of van der Waals attraction between electrode/ionic-liquid interactions on the charging processes.