  Fully compressible turbulent convection beyond the Oberbeck-Boussinesq limit
and anelastic regime is studied in three-dimensional numerical simulations.
Superadiabaticity $\epsilon$ and dissipation number $D$, which measures the
strength of stratification of adiabatic equilibria, cause two limits of
compressible convection -- nearly top-down-symmetric, strongly superadiabatic
and highly top-down-asymmetric, strongly stratified convection. Highest
turbulent Mach numbers $M_t$ follow for a symmetric blend of these two limits
which we term the fully compressible case. Particularly, the strongly
stratified convection case leads to a fluctuation-reduced top layer in the
convection zone, a strongly reduced global heat transfer, and differing
boundary layer dynamics between top and bottom. We detect this asymmetry for
growing dissipation number $D$ also in the phase plane which is spanned by the
turbulent Mach number $M_t$ and the dilatation parameter $\delta$ which relates
the dilatational velocity fluctuations to the solenoidal ones. A detailed
analysis of the different transport currents in the fully compressible energy
budget relates the low-$D$ convection cases to the standard definition of the
dimensionless Nusselt number in the Oberbeck-Boussinesq limit.
