Do we have an algorithm for comparing $e^e$ with rationals? Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence?
In a non-constructive sense, there obviously is an algorithm.

*

*If $e^e$ is some rational $q_0$, then we can decide if a rational $q$ is bigger than or smaller than or equal to $e^e$ by comparing it with $q_0$.

*If $e^e$ is irrational, then we calculate the sequences
$$z_n =\left(1+\frac{\small{1}}{n}\right)^n$$
$$a_n = \left(1+\frac{z_n}{n}\right)^n$$
$$\phantom{2+}b_n = \left(1+\frac{z_n}{n}\right)^{n+2}$$
If $a_n>q$, then $e^e>q$; and if $b_n<q$, then $e^e<q$. Since $a_n$ and $b_n$ converge to $e^e$ from below and from above, the algorithm is guaranteed to terminate with an $n$ that meets one or the other criterion.

I'd like to know an algorithm's convergence time in advance, just as for the algorithm here comparing $e^t$ and $u$ for any rationals $t$ and $u$. Do we have such an algorithm for $e^e$?
Update: We can also phrase this as comparing
$$\lim_{n\to\infty} \sqrt[n]{\frac{p}{q}n!} - \sqrt[n]{\phantom{\frac{p}{q}}\! \! \! \! \! n!}$$
with 1, since the limit is just $\log(p/q)/e$.
 A: I am pretty sure that no such algorithm is known.  Note that in particular, such an algorithm would solve the zero recognition problem for numbers of the form $e^e - p/q$ where $p$ and $q$ are positive integers. Known zero recognition algorithms for elementary constants, such as the one described in Zero tests for constants in simple scientific computations by Daniel Richardson (Math. Comp. Sci. 1 (2007), 21–37), always assume Schanuel's conjecture or some special case thereof; in your case, that amounts to assuming that $e^e$ is irrational, which is unknown.
Of course, there could still be a known zero recognition algorithm for $e^e - p/q$ since that's a very special case, but it would have to take advantage of some special feature of $e^e$ beyond the fact that it is (what I call) a closed-form number, and I don't think anything like that is known.  For example, I don't think we can even rigorously rule out the possibility that there is a sequence of rational numbers $(p_n/q_n)$ such that $|e^e - p_n/q_n| = 1/f(n)$ for some increasing function $f$ that grows faster than any computable function [EDIT: This statement is wrong; see comments below].  Of course, nobody believes such a thing, but closed-form numbers can behave worse than you might think; see for example Counterexamples to the uniformity conjecture, again by Daniel Richardson (along with Elsonbaty, Comput. Geom. 33 (2006), 58–64).
