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Ramp compression along a low-temperature adiabat offers a unique avenue to explore the physical properties of materials at the highest densities of their solid form, a region inaccessible by single shock compression. Using the National Ignition Facility and OMEGA laser facilities, copper samples were ramp compressed to peak pressures of 2.30 TPa and densities of nearly 30 g/cc, providing fundamental information regarding the compressibility and phase of copper at pressures more than 5 times greater than previously explored. Through x-ray diffraction measurements, we find that the ambient face-centered-cubic structure is preserved up to 1.15 TPa. The ramp compression equation-of-state measurements shows that there are no discontinuities in sound velocities up to 2.30 TPa, suggesting this phase is likely stable up to the peak pressures measured, as predicted by first-principal calculations. The high precision of these quasiabsolute measurements enables us to provide essential benchmarks for advanced computational studies on the behavior of dense monoatomic materials under extreme conditions that constitute a stringent test for solid-state quantum theory. We find that both density-functional theory and the stabilized jellium model, which assumes that the ionic structure can be replaced by an ionic charge distribution by constant positive-charge background, reproduces our data well. Further, our data could serve to establish new international secondary scales of pressure in the terapascal range that is becoming experimentally accessible with advanced static and dynamic compression techniques.