Abstract: The Mahler measure is defined by the geometric mean of a polynomial over the unit circle. This quantity is of importance in analysis and number theory. Answering a recent question of Granville, we show that the Mahler measure grows exponentially fast if we iterate a polynomial in the sense of complex dynamics. The exact base of that exponential growth is described by an integral over the invariant measure for the Julia set of the polynomial we iterate. We also provide sharp bounds for such integrals by using some results from complex function theory.