What are interesting examples of still uncomputed cohomology rings?

Question

It seems, that very many cohomology rings of interesting spaces have been computed. For example the integral cohomology rings of $SO(n)$, $Spin(n)$, $U(n)$, $SU(n)$, $Sp(n)$, the real and complex Grassmannians, the real and complex Stiefel manifolds are known. What are the most important examples of spaces of which the (singular) cohomology (with some coeffitients) is not known?

Answer 1

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.