  Let $\Lambda$ be the limit set of an infinite conformal iterated function
system and let $F$ denote the set of fixed points of the maps. We prove that
the box dimension of $\Lambda$ exists if and only if \[
  \overline{\dim}_{\mathrm B} F\leq \max \{\dim_{\mathrm H} \Lambda,
\underline{\dim}_{\mathrm B} F\}. \] In particular, this provides the first
examples of sets of continued fraction expansions with restricted digits for
which the box dimension does not exist.
  More generally, we establish an explicit asymptotic formula for the covering
numbers $N_r(\Lambda)$ in terms of $\dim_{\mathrm H}\Lambda$ and the covering
function $r\mapsto N_r(F)$, where $N_r(\cdot)$ denotes the least number of open
balls of radius $r$ required to cover a given set. Such finer scaling
information is necessary: in general, the lower box dimension
$\underline{\dim}_{\mathrm B} \Lambda$ is not a function of the Hausdorff
dimension of $\Lambda$ and the upper and lower box dimensions of $F$, and we
prove sharp bounds for $\underline{\dim}_{\mathrm B} \Lambda$ in terms of these
three quantities.
