Any map whatsoever from a space $X$ to $BO(n)$ gives a notion of $X$-structure for $n$-manifolds given by a choice of lift (up to homotopy) of the classifying map $M \to BO(n)$ of the tangent bundle of such a manifold $M$ to $X$ (together with a choice of homotopy).
When $X$ is itself $BG$ for some Lie group $G$ and the map $BG \to BO(n)$ is induced by a map $G \to O(n)$ of Lie groups this reproduces a more classical flavor of structure, but this more general setting includes, for example, the case of string structures, where $X = BString(n)$ does not arise in the more classical way. This very general notion of structure on a manifold arises, for example, in Lurie's discussion of the cobordism hypothesis.
So one terrible answer to your question is: there is a notion of structure on an $n$-manifold for every isomorphism class of $n$-dimensional vector bundle on some space.