What are some sets that, by construction, are known to have cardinalities larger than $\mathbb{R}$?
An example of what I'm looking for: the set of topologies on $\mathbb{R}$ has a cardinality of $2^{2^{2^{\aleph_{0}}}}$.
I don't want to involve CH or anything, just sets with beth number cardinalities such as that example.
The set of functions $\mathbb{R} \to \mathbb{R}$ is $2^{2^{\aleph_0}}$. If you consider operations like integration and differentiation to be functions on functions then the set of all these second order functions to be $2^{2^{2^{\aleph_0}}}$. You can of cause run this process of taking powersets without bound.