Models of eternal inflation predict a stochastic self-similar geometry of thermalized regions and allow existence of points that never thermalize. I explore the fractal geometry of the resulting universe in models of inflation driven by a scalar field. The fractal set of eternally inflating points is the focus of the investigation. The fractal dimension of the eternally inflating set, as well as the probability for eternal inflation to be realized, are found from suitable diffusion equations. The fractal dimension gives a bound on percolation of thermalized domains.