Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong H^*(BT)^W \hookrightarrow H^*(BT)$$ as the subring of Weyl group invariants. Except for that of $G_2$, these Weyl groups range from "large" to absurdly large, but it seems vaguely possible that these inclusions of polynomial rings might nevertheless be comprehensible. Is that the case?
Or if, more likely, explicit generators for $H^*(BG)$ wouldn't fit in this answer space, where could I find a reference for these inclusions?
NB: I am completely happy taking coefficients in $\mathbb Q$.
All of these polynomial algebras are known; this is probably very classical material but I don't know a reference. Their generators have cohomological degree $2d$ where $d$ runs over the numbers from this list. For example, $H^{\bullet}(BG_2, \mathbb{Q})$ is a polynomial algebra on generators of degrees $4$ and $12$. You might want to look up some material on Coxeter groups; IIRC there's some Coxeter group magic you can use to compute these degrees.