The charging kinetics of electric double layers (EDLs) is closely related to the performance of a wide variety of nanostructured devices including supercapacitors, electro-actuators, and electrolyte-gated transistors. While room temperature ionic liquids (RTIL) are often used as the charge carrier in these new applications, the theoretical analyses are mostly based on conventional electrokinetic theories suitable for macroscopic electrochemical phenomena in aqueous solutions. In this work, we study the charging behavior of RTIL-EDLs using a coarse-grained molecular model and constant-potential molecular dynamics (MD) simulations. In stark contrast to the predictions of conventional theories, the MD results show oscillatory variations of ionic distributions and electrochemical properties in response to the separation between electrodes. The rate of EDL charging exhibits non-monotonic behavior revealing strong electrostatic correlations in RTIL under confinement.