In this case I refer to a Hardy Field (of germs at infinity) $\mathcal{H}$ a a field of germs of real valued functions on $\mathbb{R}$ that is closed under differentiation.
That is, if $\operatorname{germ}(f)$ is in $\mathcal{H}$ then there is a function $g$ such that $\operatorname{germ}(g) = \operatorname{germ}(f)$, $g$ is differentiable and $g'$, the derivative of germ is such that $\operatorname{germ}(g')$ is in $\mathcal{H}$.
I believe it is NOT always possible to find a smooth ($\mathcal{C}^\infty$) representative of every germ in $\mathcal{H}$. If such is the case, could you provide an example? If not, a reason (proof).