Homotopy as a general organizing principle One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories comprise very general ideas that arise in a variety of places. To quote Mike Shulman: 

Personally, I don’t find it especially surprising that homotopy theory has more than one application to some other subject, any more than I would find it surprising that category theory does. I think it’s becoming increasingly clear that both of them are general organizing principles of mathematics.

I would like to build a list of interesting examples of this, especially unexpected applications of the ideas of homotopy theory in otherwise far removed areas of mathematics.
 A: Jacob Lurie just finished teaching a course at Stanford titled Tamagawa Numbers via Nonabelian Poincare Duality. You can read the definition of a Tamagawa number here (I know I'd never heard of them before). Based on that, I'm pretty surprised homotopy theory (in the guise of quasi-categories now, not model categories) comes into play here, but again it's as a unifying and organizing viewpoint which allows you to construct things purely formally via universal properties. This way of thinking has got a lot of people hopeful that Lurie's machinery can be used on all sorts of algebraic problems.
A: Having a homotopy theory for Operads gives a very general framework to talk about rectification, where you beef up the amount of algebraic structure your object has. In particular, once you realize $A_\infty$ as the cofibrant replacement of $Ass$ and once you put model structures on $A_\infty$-alg and $Ass$-alg (i.e. the full subcategory of associative algebras), then rectification is just the existence of a left Quillen functor. Similar things hold for $E_\infty$-alg and $Comm$-alg (again, if both admit a model structure) and for $L_\infty$-alg and $Lie$-alg. A good place to read about this is Berger-Moerdijk Axiomatic Homotopy for Operads (2003). The same authors later worked out the $W$-construction (or Boardman-Vogt resolution) in much more generality, which allows one to construct cofibrant replacements of operads by hand, rather than relying on general existence theorems.
A: Cardinal arithmetics is usually considered far removed from homotopy theory,
so perhaps this would be an example.
Several cardinal invariants in set theory may be viewed as derived functors, in a very degenerate model category setting.  is usually considered remote from homotopy theory.  See Gavrilovich, Hasson, A homotopy theory for set theory, Part I, Part II. 
In fact, there is also the following which appears to be an explicit attempt to 
that the ideas of homotopy theory have very broad applicability in mathematics.
Gromov, In search for a structure. Part 1, On Entropy", talks entropy in terms of category theory, but perhaps not necessarily homotopy theory. He has the following to say in a postscriptum:
 Apology to the Reader. Originally, Part 1 of ”Structures” was planned as about a half of an introduction to the main body of the text of my talk at the European Congress of Mathematics in Krak´w with the sole purpose to motivate o what would follow on ”mathematics in biology”. But it took me several months, instead of expected few days, to express apparently well understood simple things in an appropriately simple manner. Yet, I hope that I managed to convey the message: the mathematical language developed by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any meaningful idea coming from science can be fully developed in this language. Well..., actually, I planned to give examples where a new language was needed, and to suggest some possibilities. It would take me, I naively believed, a couple of months but the experience with writing this ”introduction” suggested a time coeﬃcient of order 30. I decided to postpone. 
There was also talks by Voevodsky (and older work by Tsvetkov(?)) who consider a category theory approach to probability. (There is an online 
video talk by Voevodsky on this in Russian. I'll try to recall the name of the book by Tsvetkon(??))
A: Finnur Lárusson has developed holomorphic homotopy theory in his work on Oka Theory. I really know nothing about it, but the details can be found here.
A: Another place where model categories have been used in algebra is in the study of Hopf-Galois extensions, which pop up naturally in many places, including algebraic geometry, noncommutative geometry, and Galois theory over a commutative ring rather than a field. I don't know a ton about this story, but Kathryn Hess has been working on it for years. She (with Shipley) recently developed a homotopy theory of coalgebras over a comonad precisely to apply to this type of situation. Just like with Quillen, one of the things you really want is resolutions, and to make those exist with good functorial properties you need a model category
A: Algebraic K-theory would be a huge example.  "Unexpected" is in the eye of the beholder, but I would imagine that every intrusion of homotopy theoretic ideas into the subject was unexpected by many.  
An interesting classical example is the theory of "scissors-congruence" (http://ncatlab.org/nlab/show/scissors+congruence), which can be interpreted in terms of K-theory (Zakharevich, http://arxiv.org/abs/1101.3833).
A: Over on the nLab the organizing idea is that homotopy theory and higher category are what organizes mathematics. (As in What is.... the nLab?). As a deliberate pun on Wikipedia's "neutral point of view" this got called the "n point of view".
Here "higher category theory" and the $n$™ refers $(\infty,n)$-category theory hence in particular also homotopy theory ($(\infty,0)$-category theory).
For some reason, much of the discussion we had about it was about how to talk about it without upsetting those people who don't want to believe it... In any case, we started making some pages with lists of examples. such as


*

*applications of higher category theory

*higher category theory and physics
Each time I look back at these entries I realize how imperfect they are, in spite of some time invested into them. That's how it goes. But it's a start.
A: I can't say anything about the historical context, but one of my favorite examples is topological fixed-point theory.  The fixed-point index of a smooth endomap of a finite-dimensional manifold has a purely geometric description as a sum of the indices of all fixed points.  But not only does it turn out to be homotopy-invariant, it turns out to be an instance of the general notion of symmetric monoidal trace, applied to suspension spectra in the stable homotopy category!  Moreover, this perspective makes it much easier to prove abstract properties of the fixed-point index and its various generalizations.
A: A good place to start would be the following two places where a Fields Medal was awarded to a homotopy theorist:


*

*Quillen's invention of Higher Algebraic K-theory and use of model category language to do resolutions in a context where they weren't possible before (which led to new computations, e.g. K-theory of finite fields). Quillen used model categories to prove that the derived category $D(R)$ is triangulated, a fact which had not been known previously. It also allowed him to use homotopy theoretic methods for a much wider class of rings.

*Motivic Homotopy Theory to solve the Milnor Conjecture and to reframe certain other famous number theory conjectures. This has the added benefit of allowing one to "do homotopy theory" with schemes.


Many nice references if you want to read more about the latter can be found at this MO question.
