I just remembered another family of simple groups of arbitrarily large cardinality.

Theorem (D. Lascar, 1995) Let $L/K$ be a field extension with both $L$ and $K$ algebraically closed.  Then $\operatorname{Aut}(L/K)$ is a simple group.

Since it is also true that for every algebraically closed field $K$, $|\operatorname{Aut}(K)| = 2^{|K|}$ (e.g. Theorem 80 of http://math.uga.edu/~pete/FieldTheory.pdf), it follows that for every algebraically closed field $K$ of characteristic $0$, $\operatorname{Aut}(K/\overline{\mathbb{Q}})$ is a simple group of cardinality $2^{|K|}$.  

I find this result to be interestingly reminiscent of the Schreier-Ulam-Baer theorem, and I have at times idly wondered whether there is some common explanation or a more general result that specializes to these two theorems.  (Idle indeed because I have barely looked at the proof of either theorem.)