What does "self" mean? I have had the impression that "self" has at least sometimes been thought of as "something" other than the 5 aggregates.

What does "self" mean? 

What is "sense of self"?
The fear about questions about self is misplaced. What is intended by that admonition is refraining from speculations about the origin or destiny, or existence or non-existence of a self.

What the questioner is asking, I believe, is the definition in Buddhism of Self, and that is a legitimate question to ask and answer.

The answer is that it is that which one believes is under one's control (the scope goes beyond the individual body; usually expressed as 'self or what relates to self'). And further, though it is description by negative, it is the idea that a self under one's control would not suffer pain or come to an end.

The Dhamma points out that there is only one thing that is under one's control, and that is one's acts of mind, speech or body. The outcome of one's acts, however, is out of one's control.