Extending the answer by Stefan Geschke, another example is the enveloping semigroup of a compact dynamical system. Let $G$ be a topological group that acts continuously on a compact Hausdorff space $X$. Then each element of $G$ can be identified with an element of the function space $X^X$. The closure of $G$ in this space (under the product topology) is the enveloping semigroup of $(X,G)$. These are well studied in the theory of dynamical systems, for example they can be used to prove the Auslander-Ellis theorem.

In the paper "On metrizable enveloping semigroups" by Glasner, Megrelishvili, and Uspenski (Israel J. Math 2008), there is an explicit cardinality dichotomy between $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$ in Theorem 6.1, which is related to the group $\beta\mathbb{N}$ mentioned by Stefan Geschke and by alephomega. Their preprint is at http://www.math.tau.ac.il/~glasner/papers/metr.pdf .

Extending the answer by Stefan Geschke, another example is the enveloping semigroup of a compact dynamical system. Let $G$ be a topological group that acts continuously on a compact Hausdorff space $X$. Then each element of $G$ can be identified with an element of the function space $X^X$. The closure of $G$ in this space (under the product topology) is the enveloping semigroup of $(X,G)$. These are well studied in the theory of dynamical systems, for example they can be used to prove the Auslander-Ellis theorem.

The paper "On metrizable enveloping semigroups" by Glasner, Megrelishvili, and Uspenski (Israel J. Math 2008) gives some background and restates an explicit cardinality dichotomy between $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$ in Theorem 6.1. That result is related to the group $\beta\mathbb{N}$ mentioned by Stefan Geschke and by alephomega. Their preprint is at http://www.math.tau.ac.il/~glasner/papers/metr.pdf .