Why three-phase power? Why not a higher number of phases?

In addition to PlasmaHH's answer, industry uses almost exclusively three phase power since an induction motor needs at least a three phase supply to start and run in a known direction. Single phase induction motors require lossy, unreliable, and expensive tricks to do the same (extra windings, lossy windings, speed sensitive switch, capacitors, etc).

The supply grid is based on three phase since that is the most efficient in terms of generation and delivery. Using a 9 phase grid for example would require running 9 wires for the entire distribution grid, not cost effective.

The higher order motors mentioned don't use line generated phases. Stepper motors use more phases for finer control. High order polyphase rectifiers are designed often with more 'phases', to reduce ripple, but the phases are generated locally by phase-shifting the line input by some means, either direct LC shifting, or by using a motor-generator set.

When you have single phase power distribution, you need one phase and one return, both carrying the same current.

If you now instead use symmetrical three phase power, you use three phases with a third of the current carrying capability, and you can get rid of the neutral. This simply saves some money in copper. If you now add more phases, you can not save any more copper, but only add complexity.

If you have asymmetrical three phase power, you can not get rid of the neutral, but it does not need to be able to handle all the combined current of all three phases in return. Again some copper saved. Adding more phases though will not reduce the copper needed for the neutral that much.

So yes, in the end it is more cost for virtually no gain in the average application. So you will only find more than three phases for very special things.

Three is the lowest number of phases which are equally spaced around the circle, and which can be used to create a rotating magnetic field in a given direction.

Any more phases just require more wires, and more windings in an induction motor.

Two phases can set up a rotating magnetic field if they are 90 degrees apart ("quadrature"). Quadrature-generating tricks like run capacitors are used with induction motors that run off single phase power.

Two phase power turns out not to have advantages. Motors run more smoothly on three phases, and balanced two phase requires four conductors whereas three phase requires only three. That is to say, we can link a three phase generator with a three phase induction motor using exactly three wires. Three-wire two-phase is possible, but it won't be balanced. Two of the conductors will carry the phases, and the third conductor acts as the neutral. This means that one wire has to handle more current since it is acting as a return for the other two. The three conductors under three phase all carry the same current: they are balanced.

For all these reasons, three phases represents an optimum. If it is a given that electricity is used for induction motors, more than three phases is wasteful, and so is fewer than three.

However, two-phases systems have been used, as well as higher order phase systems, like six and twelve phase, continue to be because they have some special advantages.