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.