Why Only Three Phase

Three phase is about the minimal to create a reliable rotating field that generator and motor spin at a given frequency. Also the reverse rotation function is a stable element. You can make it work with as many phases as you like, it only becomes more complicated and you will need to connect (install and pay for) more wires. Look at some stepper motors with a bunch of wires.

Old Mercury Arc rectifiers were made to work on 48 phases by the use of inter-phase transformers. How many phases do you want?

A 12 phase MAR

Imagine the complexity of getting all six phases in order to get a motor running smoothly! With three phase, the motor runs forward or reverse. To change the rotation, you change two wires! Now lets try six!! Three phase is simple, reliable and cost effective without the problems of having a phase in the wrong place! Ie the motor will run smooth in both directions!

Ye gods, reversing a motor would be a nightmare!

The "special" reason is cost, for a given amount of copper and iron we can generate and transmit the maximum amount of power using three phases. Most of the other variations of frequency, phase, number of conductors, etc. have been tried but never found widespread use in bulk power utility systems.

if you want to have a six phase system...go on...no one is stopping..but you should be responsible enough to carry the consequences there is...having all your equipments made to adopt a 6 phase system...and with all the complexity there is....try solving it out how much power will you get on a 60 degree angle...

There are 6 and 12 phase "systems" in use.

Special purpose of course.

I can think of Radar rotating motors, tethered un-manned sub power supplies to name two.

The main advantage to consider offsetting the costs described by other posters, is ripple voltage in an AC to DC converter.

With more phases, the crest factor goes down and filter capacitors banks and inductor pi filters size can be decreased.