How do models of hypercomputation overcome the Halting Problem? Hypercomputation refers to models of computation that are not possible to simulate using Turing machines.  (Hypercomputers are not necessarily physically realisable!)  Some hypercomputers have access to a resource that allows the Halting Problem for standard Turing machines to be solved.  Call this a "superpower": a hypercomputer with a superpower can decide whether any standard Turing machine terminates.


  What kinds of "superpowers" do hypercomputers use?


Ed Blakey's thesis sets up a formal framework to classify some of the major kinds of resources used in hypercomputing, but it does not try to provide a comprehensive survey of superpowers.  I am not interested in a list of hypercomputers (there is a nice list in the Wikipedia article), but in understanding what "special sauce" each model uses, perhaps thought of as a unique kind of resource.

This question is inspired by How fundamental is undecidability?.  Also related is What would it mean to disprove Church-Turing thesis? which generated lots of interesting discussion, and Are there any models of computation currently being studied with the possibility of being more powerful than Turing Machines?.
In the paper On the power of multiplication in random access machines
 it has been proven by Hartmanis and Simon that, if we add unit cost multiplication instruction in a RAM (called MRAM) then for this model P = NP. In addition the languages decided in polynomial time in the MRAM model are exactly the languages in PSPACE. 

As stated in the paper, this results shows that multiplication have the same complexity as addition iff P = PSPACE.

A more related result I have heard of, is that if we add a division instruction with infinite precision in a RAM then we can solve undecidable problems. However I could not find the paper that proves this result. If anyone is familiar with it please comment and I will update the answer.