  We obtain area growth estimates for constant mean curvature graphs in
$\mathbb{E}(\kappa,\tau)$-spaces with $\kappa\leq 0$, by finding sharp upper
bounds for the volume of geodesic balls in $\mathbb{E}(\kappa,\tau)$. We focus
on complete graphs and graphs with zero boundary values. For instance, we prove
that entire graphs in $\mathbb{E}(\kappa,\tau)$ with critical mean curvature
have at most cubic intrinsic area growth. We also obtain sharp upper bounds for
the extrinsic area growth of graphs with zero boundary values, and study
distinguished examples in detail such as invariant surfaces, $k$-noids and
ideal Scherk graphs. Finally we give a relation between height and area growth
of minimal graphs in Heisenberg space ($\kappa=0$), and prove a Collin-Krust
type estimate for such minimal graphs.
