Diffusion is pervasive in the natural world. Over one hundred years ago Einstein created a remarkably simple and effective theory describing the behavior of a single diffusing particle that has since been applied countless times to widely disparate physical systems. We will show that for large numbers of particles diffusing in the same environment, this theory does not correspond to reality as it neglects the effects of the common environment in which all particles live. This neglect leads to dramatic failures in our ability to predict the statistics of extreme diffusion, i.e. outlier particles which have moved the farthest from their starting points. We will demonstrate both theoretically and experimentally, using very different systems of colloids and optical photons, that systems with identical diffusion coefficients can have radically different outlier statistics dependent on the microscopic correlations present in the environment. We will create a new framework for the behavior of these statistics of Extreme Diffusion.