  Buoyancy-driven turbulent convection leads to a fully compressible flow with
a strong top-down asymmetry of first- and second-order statistics when the
adiabatic equilibrium profiles of temperature, density and pressure decay very
strongly across the convection layer. The growth of this asymmetry and the
formation of an increasingly thicker stabilized sublayer with a negative mean
convective heat flux at the top of the convection zone is reported here by a
series highly resolved three-dimensional direct numerical simulations beyond
the Oberbeck-Boussinesq and anelastic limits for dimensionless dissipation
numbers $0.1 \le D\le 0.8$ at fixed Rayleigh number $Ra=10^6$ and
superadiabaticity. The highly stratified compressible convection regime appears
for $D > D_{\rm crit}\approx 0.65$, when density fluctuations collapse to those
of pressure; it is characterized by an up to nearly 50\% reduced global
turbulent heat transfer and a sparse network of focused thin thermal plumes
falling through the top sublayer deep into the bulk.
