@DenisNardin For $H^2(X;G)=H^1(X;BG)$, I am more satisfied because this gives a canonical choice of "something else" to point at. Also, I am a student with quite much homotopy-theory interest too so I'd love to know more about them! Thank you.

A friend pointed out to me that $H^2$ may be of use to classify stacks, supposedly for the reason precisely the same as $H^1(X,\mathcal O^\times)$ classifying line bundles by encoding transition maps. I really don't know about this business, so if anyone could verify if this is right (and point to right references...?) I'd be very grateful.

Also, I understand that cohomology theories have very interesting relationships between them, including Serre duality, Poincare duality, Hard Lefschetz, Hodge decomposition, Frolicher relations, Lefschetz Hyperplane theorem, ... but would the intrinsic structures of $H^3, H^4, \cdots$ ever be of use to something coming from outside of the formalism?

Thank you for the comment. Could you elaborate a little more about how knowledge of higher $H^i$ can help us compute $H^0$? For example, if we didn't know $H^0$ and knew $H^3, H^4, \cdots$, then can we say something about $H^0$? Also I'd greatly appreciate some references / expository articles about how cohomology relates to syzygy, and ultimately to combinatorics!