  In this article, we further develop the thermodynamic formalism of affine
iterated function systems with countably many transformations by showing the
existence and extending earlier characterisations of the equilibrium states of
finite affine iterated function systems to the countably infinite case. As an
application, under mild conditions, we prove that the affinity dimension of a
countable affine iterated function system is equal to the supremum of the
affinity dimensions of its finite subsystems. We deduce corollaries concerning
the Hausdorff dimension of countably generated self-affine sets in dimensions
$1$, $2$, and $3$ satisfying mild deterministic assumptions and in arbitrary
dimension with generic translations.
