Both additive (CRT and screen) and subtractive (ink and dye) displays require them. But there are literally millions of subjective colors available to the human eye/brain visual processing system. Even if you just limit yourself to byte imagery, that still gives you 256 x 256 x 256 possible colors.

When we had more than 3 bands of data available, sometimes we used band combinations and assigned them arbitrarily to color guns on our CRTs. For example, if you had 7 banded Landsat data, you could use ratios or differences between two bands assigned to one color gun, so that you could fold more than three bands at a time into an image display. By subracting the red from the green, or perhaps dividing the blue by one of the infrareds, and then assigning that result to one of the RGB color guns, you could get information from more than three sensors in one image.

Sometimes (but not always) these combinations would reveal geological or vegetation features that did not normally show up. The human eye is really good at picking out color anomalies in nature--its a primate thing.