Is naturalness meaningful for non-fundamental theories? Naturalness has been a guiding philosophy for particle physics for a long time, but a few years ago I heard a talk by Nima Arkani-Hamed where he pointed out that it seems to have failed us as it relates to the Higgs boson mass and the little hierarchy problem.  He suggested that naturalness as a paradigm for particles physics may simply not be useful for understanding contemporary problems.  This got me wondering:

Has naturalness proven useful for any fields of physics other than particle physics/high energy? 

E.g. is there anything non-trivial that naturalness can tell us about condensed matter systems?
Commentary:  This may sound like a soft question, but naturalness is inherently a soft concept, yet still integral to (at least some parts of) physics.  
 A: As mentioned on the wikipedia article (note that I wrote and edited parts of that page), naturalness is arguably a specific application of Bayesian statistics. In particular, the plausibility of a model may be written as
$$
p(\text{model}|\text{data}) \propto p(\text{data}|\text{model}) = \int p(\text{data}|\text{model}, x) \, p(x|\text{model}) d^nx 
$$
where the first factor - the likelihood - measures agreement between the data and a point in the model's parameter space, and the second factor, $p(x|\text{model})$, is our prior upon the model's parameters, $x$. 
The integral, which is in fact the likelihood averaged over our prior, automatically penalizes unnatural models. If a model is 'unnatural' or 'fine-tuned', it makes bad predictions for the data in most of its parameter space. Thus, the average likelihood is small. A good model, on the other hand, predicts the data well without fine adjustment of its parameters, such that the average likelihood is big.
This of course depends on our choice of prior for the parameters. It should reflect our belief about the parameters prior to seeing the data. It is unfortunately impossible in many cases to construct a single unique prior that is compatible with our state of knowledge. As such, care must be taken about the sensitivity of the result to changes of prior within a class of priors that could reasonably reflect our state of knowledge.
Nima tells us that

naturalness as a paradigm for particles physics may simply not be
  useful for understanding contemporary problems.

This is an odd remark. If we want quantify the plausibility of contemporary models, we must use probability theory, which automatically incorporates a penalty for fine-tuning/unnaturalness. Thus naturalness is a general principle of reasoning, not an ad hoc paradigm or principle invented for particle physics that we could choose to discard if it wasn't useful.

Has naturalness proven useful for any fields of physics other than particle physics/high energy?

Since naturalness is nothing more than Bayesian model selection, and its automatic Occam's razor, the answer is obviously yes. I have no doubt that Bayesian model selection has been used in almost every branch of physics.
A: This should really be a comment, as I cannot answer the question

In physics, naturalness is the property that the dimensionless ratios between free parameters or physical constants appearing in a physical theory should take values "of order 1" and that free parameters are not fine-tuned. That is, a natural theory would have parameter ratios with values like 2.34 rather than 234000 or 0.000234.
The requirement that satisfactory theories should be "natural" in this sense is a current of thought initiated around the 1960s in particle physics. It is an aesthetic criterion, not a physical one, that arises from the seeming non-naturalness of the standard model and the broader topics of the hierarchy problem, fine-tuning, and the anthropic principle. However it does tend to suggest a possible area of weakness or future development for current theories such as the Standard Model, where some parameters vary by many orders of magnitude, and which require extensive "fine-tuning" of their current values of the models concerned. The concern is that it is not yet clear whether these seemingly exact values we currently recognize, have arisen by chance (based upon the anthropic principle or similar) or whether they arise from a more advanced theory not yet developed, in which these turn out to be expected and well-explained, because of other factors not yet part of particle physics models.

I had been an active experimental particle physicist since the 1960's and now in retirement the first time I noticed the term "naturalness" as  used  was about a year ago on a net question as this. It is obviously a value that concerns theorists developing theories. It sounds like a numerical extension to  Occam's razor

is the problem-solving principle that essentially states that "simpler solutions are more likely to be correct than complex ones." When presented with competing hypotheses to solve a problem, one should select the solution with the fewest assumptions. The idea is attributed to English Franciscan friar William of Ockham (c. 1287–1347), a scholastic philosopher and theologian.

I can understand the concern of  Nima Arkani-Hamed who is developing the amplituhedron . It certainly can  be defended by Occam's razor but probably naturalness will be a hard road :).
Certainly Occam's razor is a guiding tool in all scientific disciplines , but somebody else should answer for "naturalness".
A: It's precisely the opposite. Naturalness is more meaningful, more reliable, and less subjective the less fundamental the theory is. Naturalness arguments like those used on the Higgs mass are so common in condensed matter physics that people don't even bother to mention when they are using them.
Let me summarize what's been said already.


*

*The quantitative way to express naturalness is by Bayesian statistics. A theory is natural if it can explain observations with parameters that are likely given our priors on the distribution of parameters. This is in innisfree's answer.

*One could complain that this depends on what our prior is, so naturalness is a subjective idea. That means that the objection that something is unlikely is meaningless -- as long as it's possible, there is no problem. This is Hossenfelder's main idea and the content of Paul's answer.


It is true that what we think is natural depends on what we believe about physics in general. But that isn't a knockdown argument, that's just a description of how all science works. 
Naturalness $=$ Science
Suppose you come across an old tree in the park in mid-autumn. All the leaves have fallen off, except for one branch. All the leaves on that branch are conspicuously still there. You might come up with two theories to explain these observations.


*

*"That's just how it is." By pure coincidence every leaf on that branch has stayed on, while the others have fallen off. Somebody else might consider this unlikely, especially if they think every leaf is as good as any other, but it certainly is a possible explanation.

*You notice that branch has been grafted on. Maybe leaves from the grafted species are hardier and generally fall off later. 


If you go with theory (1), then you have basically thrown out all of science, because "unlikely but not literally impossible" is an extremely low bar for a theory to pass. If you go with theory (2), you've made progress even if you don't have a complete answer. You've at least identified something different about that branch. (This is analogous to what happens when theorists show "technical naturalness". The problem is not solved, but by establishing a symmetry, we can get a foothold that makes it easier to solve in a future theory.)
Suppose you're at the casino playing roulette. You always bet on red. The first spin is black, so you lose. The second spin is black, so you lose again. You lose $30$ times in a row without interruption. At this point you start complaining the game is fixed, but the casino manager informs you there's no solid basis for thinking that. That many losses in a row is unlikely, but not impossible. And even if you did suspect the game was unfair, the prior probability you assign to that possibility is subjective. And isn't the future fundamentally unpredictable anyway, because of the problem of induction? There's no logical reason not to keep playing forever.
Naturalness in the Standard Model
Suppose you're an experimental physicist measuring the parameters of the Standard Model. It turns out there are two angles that determine the amount of CP violation. In radians and in binary, they are 
$$\theta_1 = 1.01, \quad \theta_2 = 0.000000000000000000000000000000000000\ldots.$$
These are real measured numbers, $\theta_2$ is the theta term of QCD. There are over $30$ zeroes when it is expressed in binary, and we are finding new zeroes every few years. Model building is the act of finding hypotheses that explain this.
Even people who claim to be "above" the dirty act of model building are still doing it. To make up an example, perhaps a string theorist could say the string landscape generically results in some extremely small angles, so it's not that strange that $\theta_2$ is small. This is still model building, because (1) string theory is but an extremely complicated model, and (2) the model is being evaluated on its likelihood given a prior. (Incidentally, the Higgs mass is even harder to explain, because it's not about small numbers, but about many large numbers all adding up to almost exactly zero. You cannot fix it by just taking a prior that favors small parameter values. This distinction is usually elided in the popular literature.)
Or you might say, there's no need to postulate a specific mechanism, there's just something different about $\theta_2$ that makes the comparison to $\theta_1$ unreasonable. In that case you are still fundamentally agreeing with model builders, because you are, again, making a statement based on an inherently subjective prior. The only difference between this hypothesis and a model is that a model gives a specific reason $\theta_2$ might be different. 
The only principled way to avoid naturalness arguments is to say that there is absolutely no explanation whatsoever, now or ever, for why $\theta_2$ is small; it just is. But this is a difficult position for many to take. If you do take it, let's go to the casino.
Naturalness in Condensed Matter
Naturalness arguments work better the more you know about a subject, because your priors become more accurate. And we know very much about condensed matter at the fundamental level, because the fundamental theory is just QED, the most precisely tested physical theory in history. In some situations, we can almost compute priors semi-objectively.
Naturalness is constantly used in condensed matter physics implicitly. For example, quasiparticles can come with a "gap", the energy they have at zero momentum. Phonons are measured to be gapless, to within experimental error. This is explained by saying they are the Goldstone modes associated with translational symmetry breaking. One could also say that the many microscopic parameters describing a solid all conspired to make the gap coincidentally too small to detect, but that hypothesis is so outlandish that textbooks don't even bother to state it. The motivation behind explaining the small "gap" of the Higgs boson is exactly the same. 
The real argument against naturalness, as used in particle physics, is that we might know too little about fundamental physics for our priors to be accurate. The problem isn't that models aren't solving a real problem, but that they're so unlikely to be on the right track that trying is a waste of time. Confidence in our knowledge is a deeply personal issue with extreme variation between people. At one end, some are convinced that every problem of the Standard Model has already been solved: it's just sterile neutrinos, axions, the MSSM, a SUSY WIMP, plus a GUT. At the other end, some are convinced thinking about $\theta_2$ is pointless because we don't even know if quantum mechanics will hold up in the next experiment we do. (You could go further, to people who think we don't even know if an external world exists, but at that point you would be in the philosophy department.)
Hossenfelder's book is a statement that our priors may not be as accurate as previously believed. In that sense, almost everybody agrees with her. You will hear this constantly at talks and from rather morose reviews on arXiv. One of the original proponents of SUSY GUTs now has a cheeky plaque outside his office that declares he has "given up the search for truth". But I'm personally an optimist -- I think there is still some value to thinking about fundamental physics during the 21st century. Just as all priors are subjective, so is this attitude.
A: I can only recommend Sabine Hossenfelder's book: "Lost in Math: How Beauty Leads Physics Astray" on this topic. I think everything that needs to be said about naturalness is written down in that book. She argues that by demanding, that our fundamental theory has only natural parameters and no fine tuning, we assume to know what numbers nature is favoring.
EDIT: To better answer the question: It is questionable if naturalness is a useful concept even in fundamental physics. As every more coarse grained model of physics should be derivable from the most fundamental theories while it may lose all of the fundamental theorie's naturalness (if it exists in the first place), naturalness has nothing to say about those models. E.g. There is no reason why a condensed matter model which in principle has to be derivable from QFT should be more natural in it's parameters than QFT which already has problems with naturalness w/o SUSY.
