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There have been proposals to stop the bounties from expiring. On meta.SE these have been received rather badly, but would be most likely welcome here, as offering rewards for unsolved problems has a great tradition in mathematics, so I'm proposing to have this feature. (In case you wonder, apparently the developers monitor the meta of each site for feature-requests, so if it has a positive feedback here, it has a chance to be implemented once.)

To summarize some discussion, the main counter-argument was that such questions would occupy the feature tab for too long. I see no reason at all why the bounties couldn't be kept while removing the questions from the featured tab (possibly creating an old-bounty tab).

User yo' said that standing bounties might cause people never awarding the bounties. While I don't think this would be an issue, it could also be countered in several ways. For example, after the first 7 days, you could be offered to pay 100 more reps to keep the bounty (in the old-bounty tab), and if you ever award the bounty, you would get the 100 reps back (or optionally you could also claim it back by removing the bounty).

If you don't support this feature, I would be very interested to know why and discuss it.

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There are several sites which use a separate post on meta for collecting "standing bounties" (is there some better terminology for them)?

Of course, if MO users want to consider a similar solution, it should be discussed first. But looking at other sites, to see how it works there, might certainly be useful for such discussion.

This answer is intended as a place to collect examples of other sites using this systems. It is a community wiki answer, do not hesitate to add other examples:

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    $\begingroup$ At least one of the questions on Puzzling is a notorious unsolved math problem, the magic square of squares. $\endgroup$
    – Gerry Myerson
    Commented Sep 12, 2020 at 13:15
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    $\begingroup$ @GerryMyerson It seems that finally there's a proper incentive to solve it, then! $\endgroup$
    – Asaf Karagila Mod
    Commented Sep 15, 2020 at 0:37
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    $\begingroup$ @Asaf there's already a monetary reward on offer, multimagie.com/English/SquaresOfSquares.htm $\endgroup$
    – Gerry Myerson
    Commented Sep 15, 2020 at 8:00
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    $\begingroup$ @Gerry: But everyone knows that the real incentive to do math is fake internet points! $\endgroup$
    – Asaf Karagila Mod
    Commented Sep 15, 2020 at 8:00

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