I guess the P suggested above is power set only.

 

Assuming this I am answering the question.

 

Now we now that P(A) where A is some set is 2^|A| where |A| is cardinality of set A.

 

So P($) = 2⁰ = 1

 

So P(P($)) = 2¹ = 2

 

And P(P(P($))) = 2² = 4.

 

Thus the answer is 4.

 

Cheers