  Three-dimensionally sculpted dielectric nanostructures supporting
material(air) confined optical modes with single-emitter mode volumes
three(four) orders of magnitude below the diffraction limit, $V_{\textbf{r}_0}
\approx 0.0004 \left[\lambda/(2 n)\right]^3 \left( V_{\textbf{r}_0} \approx
0.00003 \left[\lambda/2\right]^3\right)$, are discovered using topology
optimization. Encapsulating the nanostructure by ellipsoidal shells enables
seemingly unbounded mode quality factor enhancement ($Q > 10^8$ demonstrated
numerically) leading to Purcell enhancement above $10^{11}$. It is studied how
$V_{\textbf{r}_0}$ and $Q$ depend on the choice of material, device volume,
minimum feature size and number of shells.
