  A novel concept of a \textit{mesocavity}, based on a simple yet
counter-intuitive idea of covering metal nanoparticles with unusually thick
($\gtrsim 100$ nm) high-index dielectric shells, is demonstrated in
technologically relevant and largely sought-after case of an upconversion (UC)
enhancement. For readily available combinations of materials and geometrical
parameters of metal-dielectric core-shell mesocavity, record high values of UC
enhancement of $\gtrsim 10^6$ can be achieved on the cavity surface. For
emitter inside the high-index dielectric shell with refractive index $n=2.7$,
the Lorentz local-field correction can provide an additional enhancement factor
for UC of $\approx 92$, resulting in UC enhancement of $\gtrsim 10^8$, which is
three orders of magnitude larger than it has ever been reported for individual
particles. Our mesocavity concept facilitates a beneficial synergy of plasmonic
and whispering gallery-mode resonance functionalities within an intermediary
region between nanoscale ($\lesssim 100$ nm) and microscale ($\gtrsim 1$
$\mu$m).
