Discs form a compact packing of the plane if they are interior disjoint and the graph which connects the center of mutually tangent discs is triangulated. There is only one compact packing by discs all of the same size, called hexagonal compact packing. It has been previously proven that there are exactly $9$ values of $r$ such that there exists a compact packing with discs of radius $1$ and $r$. This paper shows that there are exactly $164$ pairs $(r,s)$ such that there exists a compact packing with discs of radius $1$, $r$ and $s$. In all these $164$ cases, there exists a periodic packing.