I realise that we can't meaningfully answer the quesition "what is Planck's constant were zero" because it would break out physical models beyond recognition.
However, if we let Planck's constant approach zero asymptotically, is physics well-behaved?
For example, if you look at $E=hf$, clearly photons would carry asymptotically less and less energy, but qualitatively speaking this equation is stable. If the equation was instead something like $E=(h-\epsilon)f$ then we'd have a problem before $h$ reached zero.
So let $h$ become arbitrarily small but still greater than zero. What, if anything, breaks?
EDIT: to clarify, I am not asking what happens in the limit, I am asking if anything qualitatively changes if we take a finite step in the direction of the limit.