The most likely thing to happen is for the hydraulic press stalling out and not have enough power to push down any further or the metal in the cylinder or the press would fail and start bending.
But you are right that water, just like everything else, does compress under pressure. It is not much though and it required a lot of pressure until it gets noticeable. If you get it to extremely high pressures it will form ice which is less dense then water. And again at even higher pressures the ice changes into other crystal forms of ice. But your small hydraulic press found in most large industrial shops is not able to produce such high pressures.
Latest Answers