We studied the potential and time-dependent changes in the electric double layer (EDL) structure of an imidazolium-based room temperature ionic liquid (RTIL) electrolyte at an epitaxial graphene (EG) surface. We used in situ X-ray reflectivity (XR) to determine the EDL structure at static potentials, during cyclic voltammetry (CV) and potential step measurements. The static potential structures were also investigated with fully atomistic molecular dynamics (MD) simulations. Combined XR and MD results show that the EDL structure has alternating anion/cation layers within the first nanometer of the interface and that these structures are distinct at the most positive and negative static potentials (1.0 and -0.4 V, respectively) applied in this study. The dynamical response of the EDL to potential steps has a slow component (>10 s) and the RTIL structure shows hysteresis during CV scans (e.g., at 100 mV/s scan rate). Our results reveal that both the slow kinetics and hysteresis are due to the reorganization of the distinct EDL structures found at the extreme potentials.