Answer:
It depends on the complexity of the color wheel. You can make an accurate color wheel using only 3 colors; red, yellow, and blue. Adding green, orange, and purple can make a color wheel with 6. The standard color wheel that art students are introduced to uses these six plus six more (the tertiary colors) for a total of 12 colors.
A perfectly rendered color wheel will not have a countable number of colors. The colors will be blended into each other, and the blending will be smooth enough that you cannot differentiate where each begins and ends. You can pick out an almost infinite number of colors from the color wheel.