(1) $f:\mathbb R\to \mathbb R$ is $C^\infty$.
(2) $f^2:\mathbb R\to \mathbb R$ is $C^\infty$.
EDIT: The right formulation is: If $f\ge 0$ is $C^\infty$, then one can choose a square root of $f$ which is twice differentiable, but not better in general.
(3) $f^2$ and $f^3:\mathbb R\to \mathbb R$ are both $C^\infty$.
(4) $f^p$ and $f^q:\mathbb R\to \mathbb R$ are both $C^\infty$. EDIT: Where $p$, $q$ are relatively prime.
Obviously, (1) $\implies$ (2), (3), (4).
(2) $\implies$ $f$ is twice differentiable, in general not better.
See:
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998), p. 203-233.(pdf)
(3) $\implies$ (1). See:
MR0682456
Joris, Henri:
Une $C^\infty$-application non-immersive qui possède la propriété universelle des immersions. Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
and:
MR2179865 Myers, Robert: An elementary proof of Joris's theorem. Amer. Math. Monthly 112 (2005), no. 9, 829–831.
(4) $\implies$ (1). See:
MR0833407 Duncan, John; Krantz, Steven G.; Parks, Harold R.: Nonlinear conditions for differentiability of functions. J. Analyse Math. 45 (1985), 46–68.