gr.group theory - Naturally occuring groups with cardinality greater than the reals.

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.

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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.