Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us consider all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in all real numbers as constants). This type of structure arises in the theory of o-minimal structures, and has some rather remarkable properties. For example, every function in this class is either eventually positive, eventually zero, or eventually negative. As a result, the ring of germs at infinity forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closed field. Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain of definition contains all sufficiently large reals). Finally, these functions include all functions constructible from polynomials, $\exp$, $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise in ordinary asymptotic analysis.
Now let us apply Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for some $n, N$) to this class of functions. I claim this is a sufficient condition for (*) provided that $n, N$ are either both positive or both negative. WLOG we may assume $F$ is eventually positive. Suppose that eventually $x^n \leq F(x) \leq x^N$ where $n, N$ are both positive. By adjusting the exponents if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend to infinity. Thus, these functions are eventually increasing (their derivatives are eventually positive). By a simple derivative calculation one finds that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$
and so we get condition (*). The case where $n, N$ are both negative is similar except that we have instead
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leads to condition (*).
Edit: There are known o-minimal structures much larger than the structure I mentioned above; for instance you can expand the structure by adjoining all analytic functions whose domain is a compact box (extended to the exterior of the box by setting the function to be identically zero there).