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We have measured conductivity spectra of several supercooled monocationic and dicationic ionic liquids in the nonlinear regime by applying ac electric fields with large amplitudes up to about 180  kV/cm. Thereby, higher harmonic ac currents up to the 7th order were detected. Our results point to the existence of anomalous Wien effects in supercooled ionic liquids. Most ionic liquids studied here exhibit a conductivity-viscosity relation, which is close to the predictions of the Nernst-Einstein and Stokes-Einstein equations, as observed for classical strong electrolytes like KCl. These "strong" ionic liquids show a much stronger nonlinearity of the conductivity than classical strong electrolytes. On the other hand, the conductivity-viscosity relation of the ionic liquid [P_{6,6,6,14}][Cl] points to ion association effects. This "weak" ionic liquid shows a strength of the nonlinear effect, which is comparable to classical weak electrolytes. However, the nonlinearity increases quadratically with the field. We suggest that a theory for explaining these anomalies will have to go beyond the level of Coulomb lattice gas models.