Nanometer-sized electrodes are used to probe the transport of ions in liquid by monitoring heterogeneous electrochemical reactions. We observe pronounced nonlinearities of ion flux versus concentration when transport is localized within a region smaller than 10 nm. We show that these observations cannot be explained using conventional continuum, mean-field descriptions of ionic transport. The data indicate that these deviations are caused by the high flux of charged species that is achieved at nanometer-sized electrodes.