  We consider the growth rate of the Mahler measure in discrete dynamical
systems with the Laurent property, and in cluster algebras, and compare this
with other measures of growth. In particular, we formulate the conjecture that
the growth rate of the logarithmic Mahler measure coincides with the algebraic
entropy, which is defined in terms of degree growth. Evidence for this
conjecture is provided by exact and numerical calculations of the Mahler
measure for a family of Laurent polynomials generated by rank 2 cluster
algebras, for a recurrence of third order related to the Markoff numbers, and
for the Somos-4 recurrence. Also, for the sequence of Laurent polynomials
associated with the Kronecker quiver (the cluster algebra of affine type
$\tilde{A}_1)$ we prove a precise formula for the leading order asymptotics of
the logarithmic Mahler measure, which grows linearly.
