The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to condensed matter to high energy. Ensembles of quantum states or unitaries that reproduce the first t moments of completely random states or unitary channels (drawn from the Haar measure) are called t-designs. Entropic functions of the t-th power of a density operator are called t-entropies (e.g. Renyi and Tsallis). We reveal strong connections between the orders of designs and generalized (in particular Renyi) entropies, by showing that the Renyi-t entanglement entropies averaged over (approximate) t-designs are generically almost maximal. For random states, our results strengthen the celebrated Page's theorem by closing the complexity gap inherent in it. Moreover, we find that the min entanglement entropies become maximal for designs of an order logarithmic in the dimension of the system, which implies that they are indistinguishable from uniformly random by the entanglement spectrum. Our results relate the complexity of scrambling to the degree of randomness by Renyi entanglement entropy. The results suggest Renyi entropy as diagnostics of complexity growth beyond information scrambling/thermalization/chaos.