Ongoing “spontaneous” electrical field potentials of assemblies of neurons in the brains of diverse animal groups differ widely in character and amplitude without obvious explanation. There may be correlates with other measures of brain complexity, such as histological differentiation, but so far there are no known differences between the EEGs of humans and other mammals or between mammals and reptiles, amphibians or fish, apart from amplitude. The proposition is defended that further search for descriptors or statistical, probably nonlinear features of the time series will reveal consistent differences, meaning that we have so far missed major features of the natural history of EEGs, just as we have thus far relatively neglected the identification of features of the physiology of the brain relevant to its evolution of complexity through major grades of phyla, classes, and orders.