  We introduce and study the 1-planar packing problem: Given $k$ graphs with
$n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given
graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when
each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three
caterpillars or of two caterpillars and a path may not admit a 1-planar
packing, while two paths and a special type of caterpillar always have one. We
then study 1-planar packings with few crossings and prove that three paths
(resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen)
crossings. We finally show that a quadruple consisting of three paths and a
perfect matching with $n \geq 12$ vertices admits a 1-planar packing, while
such a packing does not exist if $n \leq 10$.
