Experiments physically performable in a finite amount of time whose results are independent of ZFC In On independence and large cardinal strength of physical statements we see that their are physical statements which are independent of ZFC, and even strong cardinal axioms. There were many answers, but unfortunately do not lead to finite experiments whose outcomes are independent of ZFC.
We can get close by running a machine that looks for contradictions in ZFC. If ZFC is inconsistent in our universe, then after (non-standard) finite amount of time, the machine will output a contradiction in ZFC. If ZFC is consistient in our universe though, this machine will never halt, and running the machine for a finite amount of time is not guaranteed to answer the question (indeed, assuming ZFC is consistient in our universe, it won't).
If we had a computer that could perform its first step in $1$ second, second step in $\frac12$ seconds, its third step in $\frac14$ steps, etc, we could perform the experiment in finite time. If ZFC is inconsistent, there will be some finite non-standard real number $r=\frac{2^{n}-1}{2^{n-1}}$ such that machine halts in the $r$ seconds (if the machine runs for $n$ steps). If ZFC is consistent, it will never halt after $r$ seconds for any $r \in [0,2)$. After $2$ seconds, we will know what the result is.
The problem is that, as far as we know, no such machine exists. Is there some other way we could devise an experiment whose result is independent of ZFC? I think it will have to involve the real numbers in some way, but I'm not sure.
 A: Based on our current partial understanding of the laws of physics, it does appear that, to the extent to which the results of physical experiments are governed by mathematical theories that we understand, they are computable, and in fact computable on a quantum computer manipulating at most $10^{122}$ bits. This means that all occurrences of real numbers in the theory are prevented from introducing incomputability.
Let me sketch how this is supposed to go. An amplitude of a particular outcome in a particular experiment might be expressed as a path integral, i.e. an integral over the infinite-dimensional space of paths a particle can take. To compute this, we could try to argue that this integral is well-approximated by an integral over a finite-dimensional space of piecewise linear paths, as long as the steps are sufficiently small. We would then argue that the integral over a finite-dimensional space could be approximated by the sum over a large number of random points in the space. By approximating the function we're integrating, we approximate the amplitude. (Of course it must be possible for physicists to computably approximate the predictions of their theories, so that they can test them in experiments. Methods like Feynman diagrams and lattice QFT enable them to do this in practice.)
Now the key point is that, whatever experiment we perform only allows us to approximate the amplitude. If we perform an experiment $n$ times, we can estimate the probabilities, and thus the amplitudes, to within an error of $~1/\sqrt{n}$. So performing the experiment $2^k$ times we only learn $k$ bits of the amplitude. Moreover, because the cosmological constant is positive, only a bounded amount of matter can fit in our observable universe, containing a bounded amount of entropy, with which we can only perform the experiment a bounded number of times before the heat death of the universe.
This phenomenon is also what shields the values of physical constants from our understandings. Only a bounded number of bits of the fine structure constant need to be known to predict every experiment that occurs in our universe.
Beyond the difficulties presented by the actual rules of the actual universe, there are a couple philosophical difficulties with your idea. The first is that, if such a theory were true, we would have difficulty testing it, as its predictions would be independent of ZFC. So what you would require is a physical phenomenon whose behavior is described by an elegant matheamtical model, which for some experiments produces predictions that can be calculated within ZFC, and for other experiments produces predictions that are independent of ZFC. Furthermore, we would have to be highly confident that no other, more mathematically sedate, theory, predicts the same behavior on the first set of experiments by a different mathematical formalism - e.g. by using whatever mathematical tools we were using to guess the answer.
I think sufficiently strong evidence for a theory of this type would indeed convince people of what you want, although there would still be dissenting views - for instance that the universe is a simulation and that the answers reflect only what the creators of the simulation think the answers to these ZFC-independent problems are, rather than the true answers. There is just no evidence I am aware of for any such theories.
A: There's an example due to Leonid Levin https://www.math.ias.edu/csdm/02-03/abstracts: 

As is well known, the absence of algorithmic solutions is no obstacle
  when the requirements do not make a solution unique. A notable example
  is generating strings of linear Kolmogorov complexity, e.g., those
  that cannot be compressed to half their length. Algorithms fail, but a
  set of dice does a perfect job!

The assertion that such a string has Kolmogorov complexity greater than half is length is almost certainly independent of any reasonable axiom system.
