There is a body of literature on the topic of supertasks, which are computational tasks involving infinitely many steps. A large part of this work involves a purely mathematical analysis and develop 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) 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 for a fun example.)
Another part of the work has considered the question of the extent to which we might actually hope to carry out such infinitary computations, perhaps taking advantage of the fact that our universe offers relativistic phenomenon that might take us beyond the Turing barrier.
(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.)
But let me strongly endorse the work of Philip Welch, who recently wrote an excellent survey describing some of the physical models in which one can compute non-computable functions by a physical procedure.