  One of Thomassen's classical results is that every planar graph of girth at
least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth
sufficiently large and a $3$-list-assignment $L$, one can do even better. Can
one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings,
or a collection of $L$-colorings that to every vertex assigns every color on
average in one third of the cases (a fractional packing)? We prove that the
packing is impossible, but two disjoint $L$-colorings are guaranteed if the
girth is at least $8$, and a fractional packing exists when the girth is at
least $6.$
  For a graph $G$, the least $k$ such that there are always $k$ disjoint proper
list-colorings whenever we have lists all of size $k$ associated to the
vertices is called the list packing number of $G$. We lower the
two-times-degeneracy upper bound for the list packing number of planar graphs
of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for
$\epsilon$-flexibility of classes of planar graphs with a given girth. For
instance, where previously Dvo\v{r}\'{a}k et al. proved that planar graphs of
girth $6$ are (weighted) $\epsilon$-flexibly $3$-choosable for an extremely
small value of $\epsilon$, we obtain the optimal value $\epsilon=\frac{1}{3}$.
Finally, we completely determine and show interesting behavior on the packing
numbers for $H$-minor-free graphs for some small graphs $H.$

Combinatorics