On the higher levels, the drills usually start with obviously distorted material but towards the end get closer to subtle saturation, even soft clipping type of processing. Start listening to similar things to what you listen in Dr. Compressor: softly clipped transients tend to be just a tiny bit rounder especially on drums. Cymbals and bass drum are good cues at least for me, they tend to sound slightly smeared. But you're right, on the higher levels it is veeery subtle.

I listen to level changes and then take note if the level is higher or lower. in 90% of cases it's higher with the distorted one. Then I listen where the distortion is, in what frequency. In some cases the level is actually lower, but during 1 level of the game it's always the same ones that are higher/lower.