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Put on a put option
I doubt that this exists, but it could theoretically. After all, a share is kind of an option to a company's future success, and so a call is already a second level on indirection. The better approach would be to 'create your own Put-Puts', by investing less money (A) in the Put you wanted to invest into, and put the smaller rest (B) in the share itself or a Call. That way, if the original Put is successful, at max (B) is lost, and if it is unsuccessful, the loss on (A) is covered by a gain on (B). Potentially, if you do the math, you can reach a mathematical equivalent situation to a Put-Put by buying the right amount and kind of Calls. However, we know already that buying a Put and a Call is a poor strategy, so that would mean a Put-Put would also be a poor strategy.