  A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets
$V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any
two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number,
$\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing
$k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily
large packing chromatic number. The question whether the packing chromatic
number of subcubic graphs is bounded appears in several papers. We answer this
question in the negative. Moreover, we show that for every fixed $k$ and $g\geq
2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the
packing chromatic number greater than $k$.
