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 and only if $K = \Bbb R(S)$. 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 are $\Bbb R(X,\sqrt{X})$ and $\Bbb R(X^{1/p^\infty}) := \Bbb R(X,X^{1/p},X^{1/p^2},X^{1/p^3},\dots)$. As you can see, even in one variable the extensions can be quite complicated. I don't know of any classification of such fields beyond the description I gave here. The answer here discusses the classification of fields in general, and what subjects are involved in trying to obtain such a classification.