In contrast to the in-plane transport electron mean-free path in graphene, the transverse mean-free path has received little attention and is often assumed to follow the "universal" mean-free path (MFP) curve broadly adopted in surface and interface science. Here we directly measure transverse electron scattering through graphene from 0 to 25 eV above the vacuum level both in reflection using low energy electron microscopy and in transmission using electronvolt transmission electron microscopy. From these data, we obtain quantitative MFPs for both elastic and inelastic scattering. Even at the lowest energies, the total MFP is just a few graphene layers and the elastic MFP oscillates with graphene layer number, both refuting the universal curve. A full theoretical calculation taking the graphene band structure into consideration agrees well with experiment, while the key experimental results are reproduced even by a simple optical toy model.