What are the known limits of the decidability of the comparison of growth rate of functions from $\mathbb{N} \to \mathbb{N}$? I am here thinking of the decidability of questions like "Is $x^x \sim 2^{\lfloor x \lg (x+2) \rfloor}$?" or "Is $2^{\lg^* x} \in O(\lg \lg x)$?".
If we restrict the functions to be polynomials (expressed in the usual manner), then this isn't hard. See also Cantor normal form.
How large can we make the class of functions before comparison becomes undecidable? Can we extend it to the functions used in a typical undergraduate algorithms class?
As Joshua Grochow explains in the comments, I'm interested really in the set of expressions, not the functions themselves. So, for instance, I'd be interested in decision procedures that could compare "$1$" and "$2$", even if they can't compare "$\ln e$" and "$n^{(\ln n)^{-1}}$".
Possibly related question: "Is the theory of asymptotic bounds finitely axiomatizable?"