From my experience playing the game, in multiple worlds, stages such as the 'collecting hearts or emblems stages' tend to have 3 crowns for completing it the first time, the Darkside in Twilight Town gives you 4 crowns and I don't think it increases beyond that point, but I haven't played Agent or Master ranked missions yet. I do remember that the Infernal Engine boss gives you 8 crowns, but I'm not sure about the other bosses or missions I have missed.
EDIT: I started a new profile just to play through the missions again and so far, Twilight Town missions 7-9 give 3 crowns and missions 11-14 give 4 crowns. If you survive the endurance mission in The Castle That Never Was, you get 4 crowns. I will either edit this post again or make a new one when I find the new amounts.