There is a body of literature on the topic of
[supertasks](http://en.wikipedia.org/wiki/Supertask), which are computational tasks involving
infinitely many steps. A large part of this work involves a
purely mathematical analysis and development of the concept,
such as my work on infinite time Turing machines ("Infinite time Turing machines," with Andy Lewis in the Journal of Symbolic Logic, 65(2):567-604, 2000, [ArXiv version](http://front.math.ucdavis.edu/9808.5093)) and
other work on higher recursion theory, E-recursion and
other infinitary models of computability. All of these
computational models exhibit functions that are not
computable by Turing machines, but are computable with
respect to the infinitary model. (See [this MO answer](http://mathoverflow.net/questions/22033/infinite-cpu-clock-rate-and-hotel-hilbert/22038#22038) for an entertaining supertask example.)

Another part of the work,however, has considered the question of the
extent to which we might actually hope to carry out such
infinitary computations. The idea is that the real universe exhibits relativistic phenomenon of which we might take advantage for computational effect. Doing so might take us beyond the Turing barrier. Hogarth and others have described physical models (Malament-Hogarth spacetimes) in which one observer has access to the results of an infinite computation carried out by another in that world. (To get the idea, imagine taking advantage of extreme relativistic time foreshortening.)

(But please beware! Although the area includes some very
interesting high-quality work, it has also attracted some
rather more questionable research, which I do not
recommend.)

Meanwhile, allow me to strongly endorse the work of [Philip
Welch](http://www.maths.bris.ac.uk/~mapdw/), who recently
wrote an [excellent survey](http://www.maths.bris.ac.uk/~mapdw/cie2007.dvi) describing some of the
physical models in which one can compute non-computable
functions by a physical procedure, among several other articles on the topic on his web page.