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?.