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.
If you look at it from the hedging perspective, if you're unsure you're going to need to hedge but want to lock in an option premium price if you do need to do so, I could see this making sense.