Are there stars with an average density greater than the central density?
$$\frac{dP}{dr} = - \rho g,$$
is the equation of hydrostatic equilibrium, where $\rho$ and $g$ are the local density and gravity, $P$ is pressure and $r$ is the radial coordinate. This can be rewritten as
$$\frac{d\rho}{dr} \frac{dP}{d\rho} = -\rho g.$$
Since $\rho$ and $g$ are positive numbers, the pressure gradient is negative.
For all types of matter $P = f(\rho)$, such that $dP/d\rho$ is also positive (i.e. pressure increases with density).
Thus $d\rho/dr$ is negative and the density increases towards the middle.
EDIT: This argument only obviously works if you ignore the dependence of pressure on other variables - such as the temperature or number of particles per unit mass in your gas. It takes more work to establish that $dP/d\rho&gt;0$ in a more general case. For example you might consider a situation where a sufficiently negative temperature gradient might counterbalance a positive density gradient.
i.e. if $P = f(\rho, T)$ then
$$ \frac{dP}{d\rho} = \frac{\partial P}{\partial \rho} + \frac{\partial P}{\partial T} \frac{dT}{d\rho}$$
and $dP/d\rho&lt;0$ if
$$ \frac{dT}{d\rho} &lt; -\frac{\partial P/\partial \rho}{\partial P/\partial T}$$
Leaving this as a placeholder for the moment.