In mathematical models, very simple communities consisting of three or more species frequently display chaotic dynamics which implies that long-term predictions of the population trajectories in time are impossible. Communities in the wild tend to be more complex, but evidence for chaotic dynamics from such communities is scarce. We used supercomputing power to test the hypothesis that chaotic dynamics become less frequent in model ecosystems when their complexity increases. We determined the dynamical stability of a universe of mathematical, nonlinear food web models with varying degrees of organizational complexity. We found that the frequency of unpredictable, chaotic dynamics increases with the number of trophic levels in a food web but decreases with the degree of complexity. Our results suggest that natural food webs possess architectural properties that may intrinsically lower the likelihood of chaotic community dynamics.