We analyze probability distributions of Birkeland current densities measured by the Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE). We find that the distributions are leptokurtic rather than normal and they are sometimes heavy-tailed. We fit q-exponential functions to the distributions and use these to estimate where the largest currents are likely to occur. The shape and scale parameters of the fitted q-exponential distribution vary with location: The scale parameter maximises for current densities with the same polarity and in the same location as the average Region 1 current, whereas the shape parameter maximises for current densities with the same polarity and in the same location as the average Region 2 current. We find that current densities |J|≥ 0.2 μA m−2 are most likely to occur in the average Region 1 current region, and second most likely to occur in the average Region 2 current region. However, for extreme currents (|J|≥ 4.0 μA m−2), we find that the most likely location is colocated with the average Region 2 current region on the dayside, at a colatitude of 18° − 22°.