It's somewhat glib to put quantum mechanics this way, but one could say that Planck's constant was zero before Planck introduced it, insofar as the classical limit of QM is $\hbar\rightarrow 0$ (which is not so very far).
Something I've played with, to absolutely no good effect so far, is the idea that quantum fluctuations might be greater or less from place to place and from time to time, which fits a little better with your Question as you have put it. However, one would then describe quantum fluctuations at different space-time points relative to the unchanged constant $\hbar$, which would now be defined as "the amplitude of quantum fluctuations under such and such conditions". That's essentially Solomoan's Answer, however his apparent assumption that the idea is obvious is a little too fast for someone with almost any philosophical leanings. [The boiling point of water is a constant, 100°C, if one defines the conditions carefully enough. A very interesting account of the historical and philosophical process associated with the definition of temperature scales is given by Hasok Chang, in a book that won a major Philosophy of Science prize, "Inventing Temperature". If you read that —it's very accessible as these things go— you will ask a different Question.]
One way to approach this Question may be to suppose that new theories introduce new constants, in terms of which old constants can be expressed, together with the conditions in which they accurately describe the Physical world. Thus, in my speculative example as I put it above, one might define the amplitude of quantum fluctuations as $\hbar$ at the surface of the earth, and describe it's variation as a function of altitude —this would at least change our theory of gravity, and likely lots else besides, so the whole of Physics becomes part of the discussion. That is a thread in Philosophy of Science that is known as the Quine-Duhem thesis, or, as Wikipedia has it, the Duhem-Quine thesis, which can be loosely summarized as the idea that a theory stands or falls as a whole, a point that I take to be vitiated by the always present possibility of changing any theory in very small ad-hoc ways.
Your Question to some extent opens a philosophical Pandora's box, which you should be cautious about opening. My view is that anyone who wants to change Physics significantly must consider these and similar ideas at length, but for Physicists trying to do the everyday job of Physics it is perhaps as counterproductive to spend time on this as it is to spend time learning to play the violin.