On page 53 of Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, Exercise 2-4 asks
How many distinct $C^\infty$ structures are there on $\mathbb{R}$? (There is only one up to diffeomorphism; that is not the question being asked.)
Is he just looking for the fact that there are infinitely many such structures, because the maximal atlases containing $x^{1/n}$ for $n$ odd are distinct? Or is there something more to say?
As mentioned in the comments you can get $|\mathbb R|$-many structures by $x \mapsto x|x|^s$, $s>0$. We want to show that this is as many as one can get. For every differential structure can be defined by a countable atlas; there are only $|\mathbb R|$-many open sets on $\mathbb R$; and there are only $|\mathbb R|$-many homeomorphisms $\mathbb R \to \mathbb R$ (they're all determined by where they send a countable dense subset). This proves the desired theorem.
This is actually true on any smoothable manifold, by roughly the same arguments.