Why are there only 3-phase electrical power systems? why not 6-phase, 9-phase, 12-phase, or some other number of phases?

A three phase system can transmit three times the power of single phase with only one extra conductor. Three phase achieves this utilization because in a balanced scenario, i.e. where each phase experiences the same load, the sum of the currents in the any two conductors at any time is equal to the current in the other conductor. Therefore no return or ground wire is required.

This maximum copper utilization cannot be achieved with a greater or lesser number of phases. Also three phase provides a rotating magnetic field that can enable an induction motor to self start. Other  phase numbers can do this as well, but three is sufficient.