Decidable theory of asymptotic growth
Hardy, in his classical book Orders of infinity, considered the class of logarithmic-exponential functions. This is a rather general class of functions, which is the minimal set of functions containing $\mathbb{R}$ and $x$, closed under addition, multiplication and division (unless it has infinitely many zeroes), closed under $\exp$ and $\log|\cdot|$ (same constraint), and closed under solution of polynomial equations (i.e. the function satisfying $f(x)^5 + f(x) = x$ is in the family). Hardy showed that any two such functions can be compared asymptotically. I'm not sure if the proof is algorithmic, but it is worth checking.

Boshernitzan extended this class even further, and doubtlessly there is other work on the subject.