  For the three-dimensional vacuum free boundary problem with physical
singularity that the sound speed is $C^{ {1}/{2}}$-H$\ddot{\rm o}$lder
continuous across the vacuum boundary of the compressible Euler equations with
damping, without any symmetry assumptions, we prove the almost global existence
of smooth solutions when the initial data are small perturbations of the
Barenblatt self-similar solutions to the corresponding porous media equations
simplified via Darcy's law. It is proved that if the initial perturbation is of
the size of $\epsilon$, then the existing time for smooth solutions is at least
of the order of $\exp(\epsilon^{-2/3})$. The key issue for the analysis is the
slow {\em sub-linear} growth of vacuum boundaries of the order of
$t^{1/(3\gamma-1)}$, where $\gamma>1$ is the adiabatic exponent for the gas.
This is in sharp contrast to the currently available global-in-time existence
theory of expanding solutions to the vacuum free boundary problems with
physical singularity of compressible Euler equations for which the expanding
rate of vacuum boundaries is linear. The results obtained in this paper is
closely related to the open question in multiple dimensions since T.-P. Liu's
construction of particular solutions in 1996 .
