The question of whether it's possible to do an infinite number of calculations has been fairly thoroughly studied, and depends on the cosmological model. Dyson 1979 (described in Baez 2004) shows that the answer is yes in an open universe with zero cosmological constant. Krauss 1999 shows that the answer is no in an open universe with nonzero cosmological constant. Neither of these cosmological models include closed, timelike curves (CTCs), i.e., there's no time travel. Since the answers depend on the cosmological parameters in ways that are complicated and not obvious even to experts, it seems unlikely to me that the answer can be answered in the form in which you presented it. To make it answerable, you would have to postulate some specific cosmological model that was compatible with current cosmological observations, but also included CTCs. The fact that the chronology protection conjecture seems to be valid suggests that you will have a hard time making any such model. There is also no reason to believe that such a model, if it did exist, would be uniquely determined by current observations.
General relativity is not generally compatible with the second law of thermodynamics. For example, there are spacetimes that are not time-orientable, and in such a spacetime there is not even any consistent way to state the second law. In a universe that was time-orientable but had CTCs, there would be no way to satisfy the second law along a CTC (since you can't keep increasing entropy forever while wrapping back around to your starting point) except in the special case where there is maximum entropy all along the CTC.
Baez, J., 2004, "The End of the Universe.", http://math.ucr.edu/home/baez/end.html
Dyson, Time without end: Physics and biology in an open universe, Reviews of Modern Physics 51 (1979), pp. 447–460, doi:10.1103/RevModPhys.51.447.
Krauss and Starkman, 1999, Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe, http://arxiv.org/abs/astro-ph/9902189