Hydrogen and hydrogen bearing compounds demonstrate or predicted to demonstrate fascinating physical properties in the limit of high densities such as insulator-to-metal transition, high-temperature superconductivity, and fluid or even superfluid ground state. These unusual behaviors are intimately connected to hydrogen quantum effects, which enhance with pressure due to an increase with compression of the zero-point energy. Knowledge of these effects is essential for understanding interiors of giant planets as well as physics and chemistry of compressed condensed matter at extremes related to possible energy applications. However, probing of hydrogen atoms remains a challenge for research at high pressures, where samples of only few micrometers in dimensions can be interrogated. In this talk, I will overview the present state of experimental investigation of hydrogen and hydrogen bearing materials at very high pressures approaching the regimes, where novel quantum phenomena become dominating. These include appearance of fluxional solid phases and insulator-to-metal transition in hydrogen, symmetrization of hydrogen bonds and superionic states in water, and formation and emergence of close to room-temperature superconductivity in poly- and super-hydrides. I will discuss the complexities of theoretical and experimental investigations at extreme conditions, which remain a challenge and continue inspiring researchers for further technical developments.