One embodiment of the present invention is a unique gas turbine engine. Another embodiment is a unique machine. A further embodiment is a unique self-aligning foil bearing system. Other embodiments include apparatuses, systems, devices, hardware, methods, and combinations for machines and self-aligning foil bearing systems. Further embodiments, forms, features, aspects, benefits, and advantages of the present application will become apparent from the description and figures provided herewith.