  A compact circle-packing $P$ of the Euclidean plane is a set of circles which
bound mutually disjoint open discs with the property that, for every circle
$S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq
P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to
both circles $S$ and $A_{i+1\mod n}.$
  We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for
which there exist a compact circle-packing of the plane consisting of circles
with radii $s$, $r$ and $1$.
  We discuss computing the exact values of such $0<s<r<1$ as roots of
polynomials and exhibit a selection of compact circle-packings consisting of
circles of three radii. We also discuss the apparent infeasibility of computing
\emph{all} these values on contemporary consumer hardware with the methods
employed in this paper.
