I am not entirely sure why you would want to do this, but if you believed that a stock would increase in value, but you doubted yourself severely, would it be possible to buy a put option where the underlying security was another put option? This seems like it should be possible, but I am not sure if it would ever be a good decision for a firm. When would a firm want to purchase an option such as this?
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.