In theoretical computer science, we classify problems according to their Turing degree. Is there any practical application of this?
Edit: Given that we cannot explicitly and mechanically understand the non-computable, what traces of non-computability (nonzero Turing degree) might we encounter in everyday life on Earth?
Any time you watch the "spinning beach ball" or "hour glass" on your computer, trying to decide whether it's time to reboot or just wait a little longer, you are doing something like trying to decide the Halting Problem which has Turing degree $\mathbf 0'$.
On the other hand, calculating the bill and the tip at a restaurant has Turing degree $\mathbf 0$.
Random sequences (almost all sequences) have Turing degree incomparable with $\mathbf 0'$.
Imagine that you conduct a "Schrödinger's Cat" type experiment repeatedly, in fact infinitely many times. Supposedly the resulting sequence of dead/alive or 0/1 bits will be random. But how random? Turing degree theory allows for a classification: will the sequence be Martin-Löf random relative to $\mathbf 0'$? Relative to $\mathbf 0''$? Relative to all reals that are first order definable in the language of set theory? What does it really mean to say that at the fundamental level, the physical world is random (i.e., follows a probability distribution)?
On the other hand, apart from collapse-of-the-wave-function phenomena, it seems that the physical world is deterministic and even computable, i.e. Turing degree $\mathbf 0$. Is this really the case, or is the appearance just a consequence of our limited abilities to understand anything noncomputable?
All this being said, experience has shown that computational complexity ($P$, $NP$, etc.) is more practically relevant than Turing degrees in many ways. And already in 1975, Baker, Gill, and Solovay showed that the methods used in Turing degree theory and computability theory more generally, being relativizable, can never settle whether $P=NP$. It kind of seems like 1975 was a watershed moment -- before that, Turing degree papers where more routinely published in Annals of Mathematics.
Focusing on applications to mathematics - as opposed to real life :P - let me mention two points I find interesting:
First, recursiveness shows up occasionally un-asked-for in classification theorems. My personal favorite example of this is Higman's Embedding Theorem, which shows that the finitely generated groups which can be embedded in a finitely presentable group are precisely the finitely generated recursively presentable groups. What makes this result interesting to me is that there is no obvious way to relativize it, leaving open the following (to me, very interesting) question: for which classes $\mathcal{D}$ of degrees is there a structural characterization of the finitely generated $\mathcal{D}$-presentable groups? (E.g., the $\Delta^0_2$-presentable groups.) Calling this an "application" is certainly a stretch, but this is one sort of area that might become interesting in the future, and I'm an optimist.
Second, there is recent work by Nabutovsky and Weinberger (see e.g. http://arxiv.org/pdf/math/9711225.pdf; also http://press.princeton.edu/titles/7903.html) using computability theory to study the structure of some complicated moduli spaces. Although the more heavy-duty results use computability theory in their statements, the simpler results involve no computability theory but have proofs relying on the fact that there is no algorithm for determining which "nice" smooth homology spheres are actualy spheres. I think in general, using computability theory to study moduli spaces could be an incredibly interesting future direction of research.