The behavior of metals at high pressure is of great importance to the fields of shock physics, geophysics, astrophysics, and nuclear materials. We study here bulk crystalline aluminum from first principles at pressures up to 2500 GPa - soon within reach of laser-based experimental facilities. Our simulations use density-functional theory and density-functional perturbation theory in the local-density and generalized-gradient approximations. Notably, the two different exchange-correlation functionals predict very similar results for the fcc -> hcp, fcc -> bcc, and hcp -> bcc transition pressures, around 175, 275, and 380GPa, respectively. In addition, our results indicate that core overlaps become noticeable only beyond pressures of 1200 GPa. From the phonon dispersions of the fee phase at increasing pressure, we predict a softening of the lowest transverse acoustic vibrational mode along the [110] direction, which corresponds to a Born instability of the fee phase around 725 GPa.