Robin, how is the transition

y = e^t -> dy/dt = e^t

=>

dy/dt = e^y -> y = -ln(C - t) -> dy/dt = 1/(C - t)

"adding one more power to the derivative in the growth equation"?

I'm not sure what that phrase you used means, exactly, but I wonder if you may be mis-visualizing the general effect of what I call "recursion".

Or what about y = t^2 => dy/dt = y^2, etc. Or y = log t => dy/dt = log y, etc.

Like I said, this doesn't necessarily hockey-stick; if you get sublinear returns the recursified version will be slower than the original.