What are interesting examples of still uncomputed cohomology rings?
People know in principle how to compute the cohomology groups of the spaces $K(S_n,1)$ — the classifying spaces of the symmetric groups $S_n$, which you can think of as the spaces of $n$-tuples of points in $\mathbb{R}^\infty$. But you'll never see them tabulated anywhere, except for some special cases, because 'in principle' doesn't mean it's easy! I wish someone would work them out more explicitly.