Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$.

The only examples I can construct are the functions $ae^{bx}+c$ for $a,b,c>0$.

Are these functions the only examples?

If not, for which nonlinear functions $g$ does $e^{g(x)}$ have this property?