The most spectacular circus perfomances were under one roof with the audience seated around three performance rings, linked in the middle with three simultaneous and somtimes diverse acts in each. Three seems a good number when you think of putting them together in a larger single circle inside the audience and not wasing too much space. It also allows good sight lines from any part of the audience to the various parts of the performance.
"Three Ring Circus" had come to mean an extravagant, skillful, and breathtaking display of all of the resources and tricks at one time. A over the top performance.