Define a greyscale picture as linear combination of its color channels¹.
Then your greyscale picture's entropy inherently is lower than a single color channel's entropy if there's an overall negative correlation between that single channel and the linear combination of the others. Entropy is the expectation of information.
I understand the use of FFT applied to gray-scale channels for this usual phenomena, but I want to know what is an instance of the opposite being true.
The FFT doesn't have anything to do with this – it just transforms channels isolatedly from a spatial base to a spatial frequency base.
¹ not all color models work like that. We can still work with that and declare the nonlinearity an error...