  We study complexity in terms of degree growth of one-component lattice
equations defined on a $3\times 3$ stencil. The equations include two in Hirota
bilinear form and the Boussinesq equations of regular, modified and Schwarzian
type. Initial values are given on a staircase or on a corner configuration and
depend linearly or rationally on a special variable, for example
$f_{n,m}=\alpha_{n,m}z+\beta_{n,m}$, in which case we count the degree in $z$
of the iterates. Known integrable cases have linear growth if only one initial
values contains $z$, and quadratic growth if all initial values contain $z$.
Even a small deformation of an integrable equation changes the degree growth
from polynomial to exponential, because the deformation will change
factorization properties and thereby prevent cancellations.
