jmc

Let $O$ be the center of the large circle, let $C$ be the center of one of the small circles, and let $\overline{OA}$ and $\overline{OB}$ be tangent to the small circle at $A$ and $B$.



By symmetry, $\angle AOB=120^{\circ }$ and $\angle AOC=60^{\circ }$. Thus $△AOC$ is a 30-60-90 degree right triangle, and $AC=1$, so $$OC=\frac{2}{\sqrt{3}}AC=\frac{2\sqrt{3}}{3}.$$If $OD$ is a radius of the large circle through $C$, then $$OD=CD+OC=1+\frac{2\sqrt{3}}{3}=\boxed{\frac{3+2\sqrt{3}}{3}}.$$