In classical electro, voltage is a continuous function of position; agreed. It is not, however, clear to me that this is true in a QM formulation. If you consider the case of a hydrogen atom, for example, the possible energies of its electron are quantised; since those energies are binding energies due to the voltage of the proton, it is not clear to me that a voltage between two allowed energy states meaningfully exists. At any rate it seems that no physical machine could output such a voltage; what would this mean? And since a physical machine ultimately consists of atoms, and sums over quantised states are themselves quantised, well then.

Given the presence of various forms of noise, I'm not sure what it would mean to measure a voltage to kilobits worth of precision. At some point I start asking how you ensure that the noise from electrons tunneling into and out of your experimental apparatus is handled. I'm also not sure that there's even a theoretical sense in which you can make your machine cold enough to reduce the noise enough to get that sort of precision.

I understand what a nine-digit voltmeter looks like. And I can extrapolate based on that, assume some things about improved materials, temperature controls, reduced temperature, extended measurement times, and so on, and would be willing to believe that the next nine digits are "mere engineering". Maybe even the nine digits after that, and the next nine. But taking that to mean you can extend it out to infinity -- literally! -- seems like a logical and physical fallacy.