In group theory, the single most important piece of information about a group is its cardinality, which is of course either finite, countably infinite, or uncountably infinite. Usually, however, uncountably infinite simply means a cardinality of $\aleph\_{1}$, the same as $\mathbb{R}$. My question is: is there anywhere that groups with cardinality strictly greater than $\aleph\_{1}$ arise naturally? Of course, it is easy enough to construct groups with arbitrarily large cardinality, but I cannot recall ever seeing them used.



Naturally occuring groups with cardinality greater than the reals.
In line with Joel's answer, my favorite "outrageously large group" is the group $G = \operatorname{Aut}(\mathbb{C})$ of field automorphisms of the complex numbers. It has cardinality $2^{2^{\aleph\_0}}$, which is pretty scary. But that's just the beginning of how large it is. For instance, from the study of real-closed fields, one can deduce that the number of conjugacy classes of order $2$ elements of $G$ is also $2^{2^{\aleph\_0}}$. It is also an extension of the absolute Galois group of $\mathbb{Q}$ (a profinite group which is conjectured to have among its quotients every finite group, up to isomorphism) by the huge simple group $\operatorname{Aut}(\mathbb{C}/\overline{\mathbb{Q}})$. 


