Specifically, is a nesting depth of more than 2 required? If so, can we determine how many are required?

This theoretical problem remained open for 25 years, to be solved by Kosaburo Hashiguchi. The answer to all questions is yes: Hashiguchi published an algorithm to determine an expression's star height in 1988.

The *generalized star height problem* is the same problem for regular expressions that include the complement operator. This is still an open problem.