One candidate for useful work is TOR-supernode work. If the internet were to become distributed and fully encrypted you'd need a lot of bogus traffic and rerout

One candidate for useful work is TOR-supernode work. If the internet were to become distributed and fully encrypted you'd need a lot of bogus traffic and rerouting so the real traffic becomes invisible. There are many obvious problems with this, but in theory the more computational resources you put into a TOR-like network the more resilient and the more anonymous it becomes.