= Solution
The notation ${}^{\mathbb R}\mathbb R$ means all functions from the <real numbers> to themselves, with no continuity restriction. Its cardinal is
$$
\boxed{\mathfrak c^{\mathfrak c}=(2^{\aleph_0})^{\mathfrak c}=2^{\aleph_0\cdot\mathfrak c}=2^{\mathfrak c}.}
$$
The exponent law follows by viewing a function to a function <set> as one function on a <Cartesian product>. <Cantor's theorem> gives $2^{\mathfrak c}>\mathfrak c$, so this is strictly larger than the other continuum-sized examples here. <Characteristic functions> of <subsets> of $\mathbb R$ already supply the lower bound $2^{\mathfrak c}$.
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