Here's the phase diagram for water:
The diagram shows us that at pressures around 1 terapascal (about 10 million atmospheres) ice is solid, at least up to 400 C. It has been predicted (reference_1, reference_2) that at higher pressures, somewhere between 1.5 and 6 terapascals, solid ice will undergo an insulator to metal transition and display properties typically associated with metals (band structure, electrical conduction, etc.). That's around 15-60 million atmospheres.
But, if the pressure was to increase towards infinity, the behaviour of water (without water being able to escape its confinement, say a ball of water at the core of a galactic mass) would be very different...
Suffice to say, it doesn't stay water after a certain point. The intense temperatures created by the compression will cause the water to break apart, eventually no longer even having oxygen atoms due to nuclear reactions. Because we're talking about an externally applied pressure, the Chandrasekhar limit doesn't apply, so there is a point at which electrons and protons combine (when the electron degeneracy pressure is overcome) and a mass of neutrons remains. Neutrons themselves also have a degeneracy pressure (though we don't have good models to predict the exact pressure that has to be overcome). From here, we don't know what happens with as much certainty, but the formation of quark matter has been predicted.
Eventually, we reach a singularity. We can think of this as all the matter we had before being compressed into an infinitesimal volume with infinite density and our applied pressure ceases to mean anything. If we started with enough water, this would behave much like any other black hole, though micro black holes are hypothesized to have some special properties.
Mentions: @ron @ Michael DM Dryden