The left-hand diagram shows the four spheres on the base. Evidently $AC = 2r\sqrt{2}$.

The right-hand diagram shows a vertical section through $A$, $C$ and the center $O$ of the top sphere. $P$ is the apex of the cone and $QR$ is a diameter of its base. Evidently $PQR$ is similar to $OAC$ and its sides are parallel and at distance $r$ outside the corresponding sides of $OAC$.

Also $AOC$ is congruent to $ABC$, so $\angle AOC = 90^\circ$. Hence $\angle QPR = 90^\circ$ also and so $OP = r\sqrt{2}$. The altitude from $O$ in $AOC$ has length $AC/2 = r\sqrt{2}$. Hence the altitude from $P$ in $PQR$ has length $OP + r\sqrt{2} + r = (2\sqrt{2} + 1)r$. Thus the radius of the base of the cone is also $(2\sqrt{2} + 1)r$ and its volume is $\frac{\pi r^3 (2\sqrt{2}+1)^3}{3}$.