Are integers unphysical/unnatural? When I first looked into number theory, I was surprised about why mathematicians are so obsessed about integers, primes and so on. As far as I understand, (almost) all integers in the real world arise as an approximate description of composite dynamics.
I think anyone here understands why individual macroscopic objects, like rocks, buildings, etc. are approximations that we can count, so I'll dive directly into quantum world.
The obvious argument about where integers could naturally arise is a quantum harmonic oscillator, which dynamics results in the energy states that we can count with integers:
$$E=(n+1/2)\hbar\omega$$
Or the energy states of atoms in the same way.
However, there are no free quantum harmonic oscillators in reals life! In real life quantum fields are interacting, and discrete energy states are simply an approximation. To see this clearly consider quantum particle. We can count individual particles, right?
No! If we take full interacting theory, the particle states polarise the vacuum and, if we start looking closer (at higher energies), we will start seeing that there is no individual particles and what we see from far away as a single particle actually is some kind of a superposition of multiparticles states. (We actually don't really know what "particles" are. Abstractly this is captured in the Haag's theorem, stating that the state space of an interacting QFT is not a Fock space.) Moreover such state will never be localized.
Looking at the world from the perspective of interacting QFT allows us to find similar arguments for any system where integers arise. Hence, integers are approximations and not natural.
The ONLY place, where I see integers as fundamental building blocks are the dimensions of symmetry groups of our world. The integer dimensions of the Lorentz group result in the 4-dimensional spacetime, the integer dimensions of the SM groups $SU(3)\times SU(2)\times U(1)$ result in the integer number of different particle types.
Update 1:
A very good point were made about the charge. If we measure the number of elementary charges in a given area, it will always be an integer number, even though the number of particles is not conserved (in QFT). It means, that any physical state is an eigenstate of a charge operator with an integer eigenvalue.
Consider the simplest case, where the minimal set of mutually commuting operators consist of the Hamiltonian, H, and the charge operator, Q. Then we can label our states in terms of their eigenvalues $|E,q\rangle$. However, are there any requirement, that $q$ is only an integer?
Update 2:
I think, I still need to elaborate on macroscopic objects to stop discussing the distribution of windows =)
Consider, say, rocks, fingers or even particles. You can count them, for example, by counting the number of distinct maxima in their density distribution. This counting is an approximation. First problem is that we need to agree what exactly will count for a single object. Second problem is that these objects are not stable in time: stones will break down and the probability distribution of a particle will become highly delocalized. The next problem is that if you zoom in, you can loose your definition of a single particle: due to vacuum polarizations any electron is surrounded by a cloud of other particles. I'm ok with integers arising from approximate description, they are just not fundamental to the theory.
Update 3:
The question was temporary closed as being opinion-based. So here are two questions that can, in principle, be answered with facts and citations:

*

*Following "Update 2", are there any mechanisms that relate the fact that the symmetry space, say $U(1)$, has integer dimensions and the fact that all the physical state are eigenvalues of the generator of $U(1)$ with integer eigenvalues? Simply put, why the charged is quantized and quantized evenly?


*Is there a way, to construct a symmetry group, such that it doesn't have an integer number of dimensions? Such that the number of generators is not an integer number (whatever that means =))
 A: Yes. Numbers are not physical objects. There’s nothing you can point to in the world and say, “That is three.” Numbers are a mathematical formalism. They’re a set of axioms that we can manipulate based on other rules and axioms (e.g., first-order logic). We then apply that formalism to the world — we notice that we can make useful predictions about objects like apples: that certain groups of apples correspond to certain integers, and that combining groups of apples corresponds to the addition operation.
But that’s true about all physics. None of it is real — it's all just formalisms that make more or less useful predictions about the world. So in classical physics, objects can be discretely counted by integers. In certain theories, particles can be counted by integers and have charges, spins, etc., measured with rational numbers. In other theories, as you describe, many things are not counted or measured with integers or rationals.
A: The number of positive charges in the nucleus of a neutral atom is always an integer, equal to the number of negative charges in its electron cloud. This fact gives rise to the entire field of chemistry, which is manifestly necessary for the existence of scientific observers.
A: 
I think anyone here understands why individual macroscopic objects, like rocks, buildings, etc. are approximations that we can count

It is the macroscopic objects that are the approximation, not the integers that you use to count them.
I have $5$ fingers on each hand. I can't define even on a macroscopic level where exactly each finger ends and my palm begins, but that does not change the fact that there are $5$ of them. Their definition is sufficient to distinguish between them enough to say that there are $5$. This is a difference between continuous and discrete measurements. Continuous measurements can never be more accurate than the definition of what they are measuring. But discrete measurements are exact by their nature, and what is being measured need only be defined well enough to distinguish between the possible values.
A: Some quantities are countable and naturally described by integers, whereas others are continuous, and consequently can be only approximated by integers - which also depend on one's choice of units.
The symmetries are the obvious place where integers appear, as you pointed out. Rotational symmetries up to six-fold symmetry are routine in crystals (although technically those with 5-fold symmetry are not crystals). The translational crystal symmetry is discrete, i.e. integers are naturally suitable for its description.
The number of particles is another example of an integers. For elementary particles it could be subsumed under exchange symmetry, but less so for composite particles, such as atoms or molecules.
I suppose that one could come with more examples where the use of integers is appropriate.
A: There are plenty of exact integers in physics.
Take a wavefunction, for example: $\psi(x) = \langle x | \psi \rangle$. You can count the stationary points and the zero crossings. 1,2,3, ....
In the room where I am sitting, there are exactly 2 windows. I know you wish to say that a macroscopic object is some sort of approximation, but you can if you like consider a density distribution of windows in my room. There will be exactly two maxima.
The energy levels of an atom do not have precise energy values owing to the interaction with the surrounding electromagnetic field. But we can still count them.
I think that integers play a large role in physics at all levels, but I agree with the observation that all our models in science are approximations and we should be alert to this.
A: First, let's distinguish two kinds of integers: immutable and mutable. Immutable integers are properties of the universe, such as the number of dimensions of space, which as far as we know cannot ever change. We can surely accept the existence of immutable integers. What's more interesting are mutable integers, such as the number of marbles in a jar, which we can use to store discrete information by taking or adding marbles. These exist but in a more tentative sense. I claim that all mutable integers are unstable and stabilizing them to long time scales requires either complete isolation from the environment or a kind of thermodynamic limit where many-body order appears.
Indeed, the fact that the marbles in a jar are roughly constant over time so long as no one touches them is an interesting physical phenomenon. The marbles neither fuse nor fiss on their own. However, this will not be the case forever. The fact is that no object is truly permanent, but that the coherence can persist on a parametrically large time scale.
As another example, consider the storage of integers in a computer. This is a physical system which contains a mutable integer whose long-time stability relies on a kind of spontaneous symmetry breaking, such as the bit stored in the magnetization of an Ising ferromagnet. Again stability requires a combination of large system size and isolation from the environment.
Andrew Steane mentioned storing an integer as the number of zeros of a wavefunction. Disregarding the difficulty of detected a zero rather than simply a tiny tiny nonzero amplitude as well as the difficulty of measuring the whole wavefunction, one has the additional difficulty of storing such a wavefunction without it becoming lost to the environment. This is like the problem quantum computerists need to solve to have quantum memory, and again this requires extreme isolation from the environment or some kind of long-range topological order where the information is stored.
Another interesting possibility is to use error correction - that is putting energy into the system to increase the lifetime of the storage. This can be done arbitrarily as long as the initial storage is good enough and there is an unlimited energy supply and place to dump entropy outside the system. Because of the second law of thermodynamics however, this will get more and more difficult over time. The point where you can use error correction is a point that classical computers have gotten to, where we don't need huge harddrives to have long term storage, and a point we're hoping that quantum computers will get to as well.
A: In Philosophy of Mathematics, integer mathematics will always be a perhaps false idealization by the way it forces itself onto the temporary physical world. Whenever you force a count of a certain item, you force an ideal identification and categorization of what you are counting.
Nowhere is this more evident than in the field of physics where you can perhaps most easily separate an object and count it as 1; astronomy. A forced definition of where that thing called a star, solar system "ends" must be applied before counting.
A: Yes, exactly half of the integers are unphysical and unnatural. You can’t physically or naturally have a negative amount of something (the rather special case where there are two kinds of property that seem opposed in some way, such as electric charge, and we have rather arbitrarily labelled them as positive and negative is more about how we label things than about a genuine physical instance of a negative quantity). There’s a reason why the natural numbers are so named.
