Levy's constant measures the exponential growth rate for the sequence of denominators of the convergents of a real number. Khintchine proved existence for almost every real number and Levy computed the constant to be $\pi^2/12\ln2$. This result is a standard exercise in modern textbooks on ergodic theory. In this talk, we generalize it to higher dimensions with Levy's constant defined using the sequence of best approximation denominators. The main ingredient of the proof is constructing the analog of the Gauss map for continued fractions. This work is joint with Nicolas Chevallier.