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.

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 realclosed 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 $\operatorname{Aut}(\mathbb{C}/\overline{\mathbb{Q}})$. 


I would expect that automorphism groups of natural structures would count as natural groups in your sense. But automorphism groups of uncountable structures often have size larger than the continuum. In general, the size of the automorphism group of a structure of size $\kappa$ is bounded above by $2^\kappa$, which is strictly larger than $\kappa$, and this upper bound is often reached, when the structure is insufficient to restrict the general nature of automorphisms. For example, the number of bijections of an infinite set of size $\kappa$ with itself is $2^\kappa$. I am sure that you will be able to construct many other natural structures of uncountable size $\kappa$, whose automorphism groups have size $2^\kappa$, and these would seem to the sort of examples you seek. P.S. Let me also note that your remark that the reals have size $\aleph_1$ is only correct when the Continuum Hypothesis holds. In general, the size of the reals, also known as the continuum, is $2^{\aleph_0}$, which is also denoted $\beth_1$, whereas $\aleph_1$ is simply the first uncountable cardinal. 


Does a group showing up in a College Algebra (precalculus) course count as arising naturally? I'm pretty sure we teach students to add two functions (from the reals to the reals) pointwise to get a new function, even there. Of course, on the one hand really we only ask them to deal with the countable subset of functions with a finite description, and on the other hand Abelian groups are not as interesting, but technically that defines a group with cardinality greater than the continuum. (We also define inverse functions and composition, but at first glance it seems that strictly monotone functions must have only continuum cardinality.) 


Any product group like $\{0, 1\}^I$ for index sets $I$, using mod 2 addition coordinatewise. It's just (isomorphic to) the power set of $I$ using symmetric difference as the addition. It's of course also a ring (pointwise multiplication / intersection ). These Boolean groups often come up in general topology. 

