Infinite time is not enough to do hypercomputation. You also need infinite memory. If you only have n bits of memory, you only have 2^n possible states, so after time 2^n, your computation must terminate or have entered a loop, so more time is not useful.
Also, your and wikipedia's description is pretty vague. Deutsch proposed a more precise model of computation along a CTC. It assumes that you only have finitely many bits of memory (and thus are computable), but it avoids the problems that shminux mentions. John Watrous and Scott Aaronson proved that in this model, you can take full advantage of the time and compute full PSPACE problems. While such problems are not supposed to be tractable in short time without a CTC, that's a far cry from halting. Also, reversible computing should make PSPACE problems practical in some sense.
"Hypercomputation" is a term coined by two philosophers, Jack Copeland and Dianne Proudfoot, to refer to allegedly computational processes that do things Turing machines are in principle incapable of doing. I'm somewhat dubious of whether any of the proposals for "hypercomputation" are really accurately described as computation, but here, I'm more interested in another question: is there any chance it's possible to build a physical device that answers questions a Turing machine cannot answer?
I've read a number of Copeland and Proudfoot's articles promoting hypercomputation, and they claim this is an open question. I have, however, seen some indications that they're wrong about this, but my knowledge of physics and computability theory isn't enough to answer this question with confidence.
Some of the ways to convince yourself that "hypercomputation" might be physically possible seem like obvious confusions, for example if you convince yourself that some physical quality is allowed to be any real number, and then notice that because some reals are non-computable, you say to yourself that if only we could measure such a non-computable quantity then we could answer questions no Turing machine could answer. Of course, the idea of doing such a measurement is physically implausible even if you could find a non-computable physical quantity in the first place. And that mistake can be sexed up in various ways, for example by talking about "analog computers" and assuming "analog" means it has components that can take any real-numbered value.
Points similar to the one I've just made exist in the literature on hypercomputation (see here and here, for example). But the critiques of hypercomputation I've found tend to focus on specific proposals. It's less clear whether there are any good general arguments in the literature that hypercomputation is physically impossible, because it would require infinite-precision measurements or something equally unlikely. It seems like it might be possible to make such an argument; I've read that the laws of physics are consiered to be computable, but I don't have a good enough understanding of what that means to tell if it entails that hypercomputation is physically impossible.
Can anyone help me out here?