Decidable theory of asymptotic growth 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?"
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.