I guess they are defining domain in a very general way. So you can say a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is taking values (as a function) in $\mathbb{R}^{\mathbb{R}}$.
Is this a common thing? I read a lot of mathematics (applied generally) and I have not seen this before, at least that I can remember.
What I have seen is this: A function $f$ on $\mathbb{R}$ is a relation such that $f :=$ {$(a,b): a \in \mathbb{R}, b \in \mathbb{R}$} $\subseteq \mathbb{R} \times \mathbb{R}$. However, as $f$ is a subset of $\mathbb{R}^2$, we interpret the function $f$ as assigning to every $a$ in $\mathbb{R}$ exactly one $b$ in $\mathbb{R}$ and write $f : \mathbb{R} \to \mathbb{R}$ such that $f(a):=b$. As mentioned in the comments, for any sets $X$ and $Y$, the set of all functions $f : X \to Y$ is denoted $Y^X$.