1) I wouldn't be so sure, that the number of stationary points and zero crossings of a wavefunctions in full interacting QFT is defined at all. If you consider a wavefunction of an electron in QED up to n-th order in perturbation theory and count the stationary points, you can add n+1 order (zoom in) and your previous calculations will become wrong. Whether we can exactly solve any real-life QFT, i don't know.

2) The distribution of windows in your room is also ambiguous, depending on where from you are measuring them. From Mars or from the point of view of an atom in your window, you wouldn't be able to distinguish whether you have 2 or 20 windows.

3) Counting the energy levels of any system is also an approximation. Consider a full state space of your system: P. Then, you identify subsets of P with respect to the energy value. A lot of possible states (superpositions) don't even have an energy value, since they are not eigenvalues of a hamiltonian. Say now you want to count, how many subsets of P consist of eigenstates and have different energy values. I an not sure such subsets are closed or even existent in full interacting QFT.

@VladSamoilov I think your comments are all worthwhile and interesting points about QFT. But don't forget that QFT must reproduce everyday observations, such as the output of experiments measuring line spectra. Although no line spectrum is perfectly resolved, many of the peaks are countable (as are the clicks in particle detectors).

@VladSamoilov The distribution of windows still exists even if it's hard to measure accurately from Mars.

@user253751 But windows, as well as stones or any other macroscopic objects are composite. I'm ok with integers arising approximately through counting extrema of density distribution. They are not fundamental, though.

How about the number of dimensions? Are there any theories that posit a non-integral number of dimensions? Yes, there are fractals but these are a mathematical construct with just approximate realisations (e.g. coastlines).

@badjohn A number of dimensions is a number of numbers that you need to describe a position, but numbers are unnatural. Or the number of perpendicular lines you can make, but lines are unnatural and so is perpendicularity...

@user253751 How about the power in Newton's law of gravity? Another example is the $2$ in the formula for kinetic energy or Einstein's famous equation. Are these approximations?

@badjohn Newton's law is very much an approximation, but that is a good point in general. I though about how to describe the dynamics without integers. If you have a full state (phase) space of your system, your equations of motion just tell you how the neighboring values are related. These equations are secondary to the state space and are often an approximation.

@VladSamoilov How many dimensions does spacetime have?

@badjohn Strange question, but 3+1.

@VladSamoilov We use integers in physics because many things in nature come in discrete quantities. We can count the number of radioactive decays made by a lump of radioactive material with a long half-life. We can physically count the number of ions we have in our ion trap. We can count the number of times a cosmic-ray muon passes through our detector.

@VladSamoilov But your answer was an integer. Is that just an approximation?

@badjohn Please, read the question. I stated that the only places I found integers are the dimensionalities of symmetry groups. And the question was, whether they are approximations, too.

@badjohn aliens define energy as double our definition so they don't have the factor of 1/2. Powers in those formulas are unnatural, in nature it's just the thing times the thing.

@user253751 multiplying is unnatural, in nature it's just repeated sums