Every public goods problem is mathematical research is a case where the solutions are socially useful but not saleable. The problem (pointed out by the sibling) with primecoin was that it worked too well, such that further solutions were uninteresting and past the point of diminishing returns.

Edit: See my here for a model of how NP complete problems could fill the role of you could predict difficulty in advance. There wouldn't be double dipping because the person wanting the answer wouldn't pay both bounties.

https://news.ycombinator.com/item?id=15107653