Covering $\mathbb R$ with translates of a compact set

Motivated by a question of Gruenhage, we investigate when $\mathbb R$is the union of less than continuum many translates of a compact set $C \subseteq\mathbb R$. It will follow from one of our general results that if a compact set $C$ has packing dimension less than 1, then $\mathbb R$ is not the union of less than continuum many translates of $C$.