The Birkhoff Ergodic Theorem can be seen as a dynamical version of the law of large numbers. A natural question is how quickly the measure of the exceptional set decays; this leads to the study of large deviations. In this talk, we present large deviation estimates for diagonal actions on the space of lattices and discuss an application to the effective uniqueness of the measure of maximal entropy. This is an ongoing joint work with Elon Lindenstrauss and Ron Mor.