  We study the growth of degrees in many autonomous and non-autonomous lattice
equations defined by quad rules with corner boundary values, some of which are
known to be integrable by other characterisations. Subject to an enabling
conjecture, we prove polynomial growth for a large class of equations which
includes the Adler-Bobenko-Suris equations and Viallet's $Q_V$ and its
non-autonomous generalization. Our technique is to determine the ambient degree
growth of the projective version of the lattice equations and to conjecture the
growth of their common factors at each lattice vertex, allowing the true degree
growth to be found. The resulting degrees satisfy a linear partial difference
equation which is universal, i.e. the same for all the integrable lattice
equations considered. When we take periodic reductions of these equations,
which includes staircase initial conditions, we obtain from this linear partial
difference equation an ordinary difference equation for degrees that implies
quadratic or linear degree growth. We also study growth of degree of several
non-integrable lattice equations. Exponential growth of degrees of these
equations, and their mapping reductions, is also proved subject to a
conjecture.
