Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In order to tie up a loose end over at this question, I wonder
Do there exists arbitrarily large simple acyclic groups?
More generally, do there exist arbitrarily large simple groups $G$ such that there exists an acyclic space $X(G)$ with $\pi_1(X(G)) = G$?
Do there exist arbitrarily large simple groups $G$ with $H_2(G; \mathbb Z) = 0$ -- or equivalently (I think) for which there are no nontrivial central extensions?
Heck, what is one example of a simple nonabelian group $G$ with $H_2(G;\mathbb Z) = 0$?
(2) is all I really need, for which (3) will suffice (see below); (1) is just a natural strengthening.
There is a proper class of simple groups; e.g. the alternating group on any set is simple (though not acyclic).
There are also acyclic spaces with arbitrarily large fundamental group, cf. Kan-Thurston, but the constructions I've seen don't produce spaces with simple fundamental group.
In the comments at the above-linked question, Tom Goodwillie points out that a positive answer to (3) implies a positive answer to (2) by taking $X(G)$ to be the fiber of $BG \to BG^+$.
I've included the "model theory" and "logic" tags mostly because I suspect maybe the people who know the most about very large simple groups might just be logicians. But if these tags seem inappropriate, I wouldn't object too strongly to removing them.