The many worlds interpretation (MWI) takes quantum mechanics (QM) seriously as a description of how the world works.
A measurement is an interaction that takes information from a system that can be copied into other systems. This copyable information for a particular interaction consists of eigenvalues of some observable:
If information about some observable is copied widely then that prevents interference between versions of the system with different values of that observable. The suppression of interference makes these different versions act like a collection of parallel universes to a good approximation:
The set of values of an observable form a discrete and finite set for any finite system. So the number of distinguishable versions of a system is some finite but large number. To calculate the number of universes that could be produced you would do something like the calculations of entropy in "Universal upper bound on the entropy-to-energy ratio for bounded systems":
This number isn't particularly interesting or fundamental and it doesn't really matter for making predictions, so there's not much reason to calculate it.
Now, if we look at some particular member of the set of measurable values, it has a real valued number that represents the probability of getting that result. So each branch consists of a set of instances of a system that aren't distinguishable by any measurement. The "number" of instances is either a real number or a continuous infinity depending on how you want to think about it.