Due to their straightforward design and advantageous operational characteristics, cascaded arrangements of overdriven free-displacer thermocompressors may provide a simple, expedient and cost-effective solution to reduce the high primary energy requirements for compressed air generation, which is a common challenge in industry. Following first experiments with a single-stage prototype, this contribution presents the first experimental realization of a three-stage cascade so that the overall concept has now been demonstrated to be fully operational for the first time. In particular, the predicted self-control capabilities of a stage located within the cascade and thus without any preset inlet or outlet pressure could be proven, whereas this had been impossible with the single-stage prototype before. Stable operation with all stages running is possible at total pressure ratios ranging from a practical maximum near 2.1 down to approximately 1.36. At even lower values, the pressure ratio of the first stage undercuts an analytically predicted lower stability limit and eventually stops, whereas the power density of the remaining stages is increased. To exploit the potential for improvement identified during the single-stage experiments, the design of the new stages is based on the first prototype, but has been optimized by a similarity-based scaling procedure and design evolution, resulting in a significant increase in power density from 7.1 W to 12.5 W per liter of displacement. To further demonstrate the capability of such a cascade to extract waste heat from a hot stream, the heater temperatures of the stages were successively reduced along the cascade by an adequate control, thus emulating a finite heat capacity rate of the stream. Although the exergy transferred to the compressed air flow is reduced in this case, the overall performance is not adversely affected. The combination of the models and the scaling approach, which have now been demonstrated experimentally, provides a comprehensive toolbox for the future development of a series machine.