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If A = The Empty Set, $\emptyset = \{\}$ , then

$\mathcal{P}(\mathcal{P}(A)) =\{ \color{#D61F06}{\{\}}, \color{#3D99F6}{\{\{\}\}}\}$

The power set of the empty set is not the empty set, it is a set which contains one element, the empty set. The power set of that set is a set with two elements: the empty set, and a set which contains the empty set. So the answer is 2.

$|\mathcal{P}(\mathcal{P}(A))|= 2$

Simply use $\left|\mathcal{P}(X)\right|=2^{|X|}.$ In this case $|\mathcal{P}(\emptyset)|=2^0=1$ and then $|\mathcal{P}(\mathcal{P}(\emptyset))|=2^1=2$ .