How pure are the colors of the rainbow? I just watched this lecture on rainbows https://www.youtube.com/watch?v=iKUSWJWMSk4 and in it I found out that the colors in the rainbow are not monochromatic. At any point in a rainbow there is a peak light frequency, but all the lower frequencies are also present. So the red side of the rainbow is nearly monochromatic but the blue side is mixed with white light. My question is how many times more intense is the blue light than the white light it is mixed with?
The reason I want to know this is because I want to know which visible colors can't be made by mixing colors of the rainbow. If the blue light is say 10 times stronger than the light it is mixed with, then there's probably only a tiny sliver of visible colors that can't be made from a rainbow.
Another reason why the colors aren't perfectly monochromatic is that the intensity peaks aren't perfectly sharp, but I'm guessing this is a less significant factor.
 A: Color is a double valued concept, different in physics and in perception.
In Physics there is a one to one correspondence between color seen in the visible spectrum and the frequency of light.


Electromagnetic spectrum with visible light highlighted

The whole electromagnetic spectrum covers many frequencies above an below the visible, which are the colors seen in rainbows.
In biology and perception due to biology perceived colors are mixtures of frequencies.


The reason I want to know this is because I want to know which visible colors can't be made by mixing colors of the rainbow

You are talking of the perception of color, not of the color defined by the frequencies, and it is , as seen above, a more complicated effect. See also my answer here.
Rainbows in the air will depend on a number of properties of the atmosphere, and though I have no time to see the video you link, there will be  effects dependent on the atmosphere and the particulars of the scattering of light . But the mixing color effect you are looking for has to be found in the biological observation of color.
