  We show that for any $m\in\NN\cup\{\infty\}$ there exist $m$ disjoint FB
domains whose union is dense in $\CC^k$. In fact we show that any point not in
the union is a boundary point for all the domains. We construct FB domains that
contains arbitrary countable collections of subvarieties of $\CC^k$, and we
construct FB domains that intersect elements of countable collections of affine
subspaces of $\CC^k$ in connected proper subsets. Moreover, we show that any
Runge FB domain is the attracting basin for a sequence of automorphisms of
$\CC^k$, although not necessarily if you only allow iteration of one
automorphism. We also show that an increasing sequence of Runge $\CC^k$'s is a
$\CC^k$.
