On the cusp of the 100 year anniversary, Khinchin’s theorem implies a strong 0-1 law for the real line; namely, the set of real numbers within distance q^{-2-\epsilon} of infinitely many rationals p/q is Lebesgue measure 0 for \epsilon>0, and full measure for \epsilon=0. In these lectures, I will present an analogous Khinchin-type theorem for horoballs packings in Gromov hyperbolic metric spaces that arise from geometrically finite actions with controlled parabolic growth. An application is the logarithm law for cusp excursion; that is, we prove asymptotics for the depth in the horoball packing of a typical geodesic. This is joint work with Giulio Tiozzo.