To which character or characters does a Kleene star apply? If you have a Kleene star applying to a set of characters not in any closure, does it apply to that whole string, or just the one character it belongs to? Any examples I search don't specify.

For example:

(A)BB*

Does this mean 1: one A followed by zero to an infinite number of BB's, or does it mean 2: one A followed by at least one B and zero to infinite number of B's?
Kleene stars applies to the character(s) it belongs, so in your example the correct option is the 2° one.

ABB* = {AB, ABB, ABBB, ABBBB, ..}

A(BB)* = {A, ABB, ABBBB, ABBBBBB, ..}