We address the implications of finite ionic size and solvent polarization on the response of the electric double layer (EDL) at two cation-selective electrodes in nonequilibrium conditions. The current between the electrodes is driven by a steady-state dc bias in conjunction with a probing high-frequency ac voltage. We report that the finite ionic size (steric) effect is prominent at high voltages near the electrodes where the ion densities are high, while the solvent polarization dramatically alters impedance characteristics for thick EDLs owing to the alteration of solvent permittivity in regions with a high electric field. Depending on the magnitude of the dc bias, our results show that the steric effects and solvent polarization lead to dramatic alterations in the net impedance for moderately thick electric double layers as compared to an extremely thin one. We also highlight that the solvent polarization suppresses the anomalous growth of dc current (anomalous rectification effect) for applied high-frequency ac voltages.