It actually can happen, but it's rare.

I'm assuming you already know how rainbows are formed by light refracting through droplets of water. Double rainbows (which I'm going to call second-order rainbows here) happen when some of the light refracts immediately but some of it bounces around inside the droplet before escaping. It has to bounce twice to produce a double rainbow, amd because some of the light is lost with each bounce, the second rainbow is fainter than the first.

Higher-order rainbows are possible, and may not even be all that rare. The problem is that each order requires more bounces than the last, and because each bounce loses some light, the higher-order rainbows are harder to see. The first photograph of a triple rainbow in nature was only made in 2011, because the light has to be very bright and the sky has to be very clear. At least one person has managed to photograph a quintuple rainbow (fifth-order). This was done in 2014.

In laboratories, of course, it's possible to use much brighter lights and better conditions than those found in nature. In the mid-1800s, Felix Billet created up to 19th-order rainbows in his experiments. In 1998, scientists using lasers created up to 200th-order rainbows. What we weren't sure of until recently was whether you could get the right conditions to see more than two in nature. But it turns out that you can.