*2. A sphere and a cylinder are on the same plane, and they have a common inscribed sphere. Let the volume of the sphere be $V_{1}$, the volume of the cylinder be $V_{2}$, and $V_{1}=k V_{3}$. Then the minimum value of $k$ is
2. $\frac{4}{3}$.

Take their common axial section, as shown in the figure, then $r=r_{2}$, $r=r_{1} \tan \theta, h=r_{1} \tan 2 \theta$.
If $V_{1}=k V_{2}$, then we have
$$
\begin{array}{l}
\frac{1}{3} \pi r_{1}^{2} \cdot r_{1} \tan 2 \theta \\
=2 \pi r_{1}^{3} \cdot k \tan^{3} \theta .
\end{array}
$$

Simplifying, we get $3 k^{4} \tan^{4} \theta-3 k \tan^{2} \theta+1=0$.
Since $\tan \theta$ is a real number, we have $\Delta=9 k^{2}-12 k \geqslant 0$.
But $k>0$, so $k \geqslant \frac{4}{3}$. When $k=\frac{4}{3}$, we get $\tan \theta=\frac{\sqrt{2}}{2}$, which makes the original problem meaningful. Therefore, the minimum value of $k$ is $\frac{4}{3}$.