We prove the existence of a global attractor in ${\mathcal{S}}_0^2(\mathrm{\Omega })\cap L^{2p-2}(\mathrm{\Omega })$ for a semilinear strongly degenerate parabolic equation in a bounded domain with the homogeneous Dirichlet boundary condition, in which the nonlinearity satisfies a polynomial type condition of arbitrary order and the external force belongs to $L^2(\mathrm{\Omega })$. This global attractor is then shown to have a finite fractal dimension in $L^2(\mathrm{\Omega })$. We also study the existence and exponential stability of the unique stationary solution to the problem.