Purely transcendental extensions of $\mathbb{R}$ are those of the form $\mathbb{R}((X_i)_{i \in I})$ where $I$ is a set and the $X_i$'s are (distinct) indeterminates.
Now, I wonder if there is a transcendental extension of $\mathbb{R}$ that is not purely transcendental.
If such extensions exist, I wonder if there is a classical way to classify them in different categories.
If $K/\Bbb R$ is a field extension, you can always find a transcendence basis $S$; i.e., a set $S\subseteq K$ such that $K$ is algebraic over $\Bbb R(S)$. The extension is purely algebraic if there exists a subset $S\subseteq K$ which is algebraically independent over $\Bbb{R}$ and $K = \Bbb R(S)$. So you're asking for a classification of algebraic extensions of $\Bbb R((X_i)_{i\in I})$. As I mentioned in the comments, $\Bbb C(X)$ is such an extension, but so is something like $\Bbb R(X^{1/p^\infty}) := \Bbb R(X,X^{1/p},X^{1/p^2},X^{1/p^3},\dots).$ As you can see, even in one variable the extensions can be quite complicated.
The answer here discusses the classification of fields in general, and what subjects are involved in trying to obtain such a classification.