Let \(P(X)\) denote power set of a set \(X\). If \(A\) is the null set, then the number of elements in \(P(P(P(P(A))))\) is

The fourth power set contains $16$ elements. A set of size $n$ has $2^n$ subsets, so four successive applications starting at size $0$ give $1,2,4,16$. A count of $15$ omits the empty subset from the final step.