In a typical probability theory course, all of them, hopefully. I found random variables to be a crucial part of my probability theory class, being used in almost all (if not all) topics after the first month.

There are certainly other approaches to learning probability theory (though perhaps not what you were thinking.) I understand that E.T. Jaynes' book on probability theory, which I have not yet read, does not go the route of random variables.

They may use random variables frequently without being able to define them.

Jaynes' book is excellent but is more about the philosophy of the Bayesian interpretation of probability than mathematics.

His problem with "random variables" is not so much the concept (although he wasn't a fan of measure theory), but the "random" in the name.

This blog entry is meant for a review for a graduate level course, so it makes sense that it runs through a lot of formal definitions.