Orange and brown exist. Oranges are the various wavelengths of light between red and yellow, and browns are just those wavelengths with lower saturation.

Purple doesn't exist in the sense that there's no pure wavelength of light we can ever associate with it. (Caveat here that indigo/violet, which do have pure wavelengths, often get colloquially lumped in with purple/magenta.) Purple/magenta is just an optical illusion resulting from how our biology works. In a gross oversimplification of what is actually happening, our brains take the RBG signals we receive and sort of average them out. Since both the red and blue sensors are being triggered our brain would ordinarily average that out to be green since that's the wavelength of light that lies between the two, but our green sensors aren't being triggered at all so our brain makes up the magenta/purple colour to be able to process that conflict. This has the illusionary effect of linking the red and blue ends of the physically linear colour spectrum into a perceptually circular one.