Maximizing a monotone supermodular function s.t. cardinality
I think this is an example showing no kind of approximation is possible except with exponential$(k)$ value queries.

Let $f(S) = 0$ if $|S| \leq k$, otherwise $f(S) = |S| - k$. Now pick a special set $S^*$ uniformly at random from all sets of size $k$, and let $f(S^*) = 0.5$.

I'm claiming that this function is supermodular because every element initially has $0$ marginal value, then possibly $0.5$, then $1$ thereafter. So marginal value of any element only increases in a superset.

$S^*$ maximizes the function subject to a cardinality constraint of $k$, with an objective value of $0.5$.

Meanwhile any algorithm making subexponentially many value queries will be unable (except with tiny probability) to find a feasible set with nonzero value.