Infectious disease outbreaks and other contagion, e.g., memes, recapitulate biology: they emerge from the multi-level interaction of hosts, pathogens, and their shared environment.  Here, by studying the predictability of a diverse collection of historical outbreaks--including, chlamydia, gonorrhea, hepatitis A, influenza, dengue, measles, polio, whooping cough, Ebola, and Zika--we identified a fundamental entropy barrier for contagions.  We further demonstrate that social network heterogeneity is the most likely explanation for observed differences in entropy across contagions.  Our results highlight the importance of dynamic modeling approaches for prediction and hypothesis testing.