I think it would ultimately depend on a use case for that metric, otherwise you're putting the cart before the horse. There are many measurements and calculations you could come up with, but no obvious (to me, anyway) interpretation of "most discontinuous": something is either in one piece or not. If you needed a metric like this for a practical purpose, your specific needs would be a starting point for designing one. If it's more of a shower thought, you sort of have "too much freedom" to be able to define anything that's necessarily meaningful.
Simple examples would be just "number of 'discrete parts'", "minimal area needed to span all territories" and things like that. Maybe you're more interested in "total distance from all satellites to wherever the capital is" or something, in a different context. The point is they'd all tell you radically different things, so it's important to know which one to ask for.
You could argue that something like Hawaii and Alaska's distance from the rest of the US makes the US score highly.
You could argue that any number of island nations score highly because after all, most of e.g. the US is in one part.
You could argue e.g. Norway's territories near both poles make it pretty high-scoring too.
You could argue that for whatever reason, distribution of area and population matter, and so on.