Complete integrability in a symplectic set-up means the existence of a Lagrangian foliation, leaf-wise preserved by the dynamics. We shall describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This structure implies the existence of an invariant contact form, providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. An example of contact complete integrability will be presented: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. A surprising acceleration mechanism for closed light-like billiard trajectories will be described.