Hydrogen gas compression systems that each include a multistage centrifugal compressor in which each stage has an inlet-to-outlet pressure rise ratio of about 1.20 or greater. In one embodiment, the multistage compressor includes six high-speed centrifugal compressors driven at a speed of about 60,000 rpm. The compressor has an output of more than 200,000 kg/day at a pressure of more than 1,000 psig. The compressors for the compression stages are distributed on both sides of a common gearbox, which has gearing that allows axial thrusts from the compressors to be handled effectively. Each stage's compressor has a unique impeller, which is secured to a support shaft using a tension-rod-based attachment system. In another embodiment, the multistage compressor is driven by a combustion turbine and one or more intercoolers are provided between compression stages. Each intercooler is cooled by coolant from an absorption chiller utilizing exhaust gas from the combustion turbine.