You might be tempted to say "lots". And you would be right, as far as that goes.
But that doesn't satisfy a real mathematician. The question that immediately arises is whether "lots" is "enough". And that leads the better sort of mathematician inevitably to: "enough for what?" That is what mathematicians are deep in the middle of exploring, now.
For example, when you are asked, "Does this skirt make my butt look too big?", you obviously must not say "yes", but you just as obviously cannot, with an entirely clear conscience, say "no", either. But for anyone with an intact survival instinct, the counter-question, "too big for what?" should spring immediately to mind. And it's a good one, but it depends intimately on the true size of the set of real numbers. So, this is not an idle pursuit.