  We develop a thermodynamic formalism for quasi-multiplicative potentials on a
countable symbolic space and apply these results to the dimension theory of
infinitely generated self-affine sets. The first application is a
generalisation of Falconer's dimension formula to include typical infinitely
generated self-affine sets and show the existence of an ergodic invariant
measure of full dimension whenever the pressure function has a root.
Considering the multifractal analysis of Birkhoff averages of general
potentials $\Phi$ taking values in $\R^{\N}$, we give a formula for the
Hausdorff dimension of $J_\Phi(\alpha)$, the $\alpha$-level set of the Birkhoff
average, on a typical infinitely generated self-affine set. We also show that
for bounded potentials $\Phi$, the Hausdorff dimension of $J_\Phi(\alpha)$ is
given by the maximum of the critical value for the pressure and the supremum of
Lyapunov dimensions of invariant measures $\mu$ for which
$\int\Phi\,d\mu=\alpha$. Our multifractal results are new in both the finitely
generated and the infinitely generated setting.
