Of all univariate distributions on the nonnegative reals of a given mean, the distribution that maximizes the Rényi entropy is Lomax. But the memoryless Lomax stochastic process does not maximize the Rényi entropy rate: For Rényi orders smaller than one the supremum of the Rényi entropy rates is infinite, and for orders larger than one it is the differential Shannon entropy of the exponential distribution, which is the distribution that maximizes the differential Shannon entropy subject to these constraints. This is shown to be a special case of a much more general principle.