Dear Neil,
as you surely know, some interpretations of QM take an analog and deterministically evolving wave function as a fact. For example DI or MWI. I pondered about wether those views are really a fact or not.
To "answer" the contest's question more definitely, one has to provide an experimentum crucis that could differenciate between a strictly analog model of reality and other (non-strictly) models. This task seems to be somewhat a variation of the halting problem, because if you can measure some values to - say - the 70 billionth decimal place, that's in no way a proof that nature acts in an infinitely precise manner or not. Even if the next 50 million decimal places are only zeros, you have no garantuee that after that there will again exlusively only follow further zeros and no other numbers.
If we define "information" as something that made a factual distinction in the past (due to measurement-outcomes or mere interactions of particles), then at least within our observable horizon of the universe there should have been produced only a finite amount of information - resulting in our present configuration of the observable horizon. Though "observable" means "factual", we cannot exclude the possibility that unobserved, "counterfactual" information is preserved via an analog and deterministically evolving wave function, be it as many worlds or as other unobservable dimensions.
If the holographic principle is valid, the total information content of a region of space is finite. For our observable universe, this amount of information - due to the holographic principle and the calculations of Seth Lloyd and others - cannot exceed about 10123 factual information units (bits). So, if nature would be able to outdistance this number by a quantum computer's operation that leads to a factual, verifiable output (in my experiment via the factorization of large numbers into their prime components), this would indicate - at least - that the underlying wave function does operate in a counterfactual realm, a realm that cannot be associated with ordinary space and time.
But if nature wouldn't be able to outdistance the 10123 factual bit-flips, this would be - in my opinion - a strong hint for reconsidering a strictly analog and deterministic view of QM.
"Are those particles really used, and would our doing an experiment right now actually mesh with the potential capability of all those entities in principle?"
I really don't know if those particles would be really used in such an experiment, but if the claims of the finite information bound (or let's say, the finite computational capability) of our observable universe is valid (independent of how the universe computes or is interconnected in detail), there should indeed occur a breakdown of the "wave function" at some critical point of my proposed experiment.