Let $S$ denote the set of all functions $f: \{0,1\}^4 \to \{0,1\}$. Denote by $N$ the number of functions from $S$ to the set $\{0,1\}$. The value of $\log_2 \log_2 N$ is _______.

1. The domain of each function $f$ is $\{0, 1\}^4$, which has $|\{0, 1\}^4| = 2^4 = 16$ elements.

2. The codomain is $\{0, 1\}$, with cardinality $2$.

3. The total number of functions from $\{0, 1\}^4$ to $\{0, 1\}$ is:

   $$|S| = 2^{16}$$

4. $N$ is the number of functions from $S$ to $\{0, 1\}$. Since $|S| = 2^{16}$ and the codomain has size $2$, we have:

   $$N = 2^{|S|} = 2^{2^{16}}$$

5. Now calculate $\log_2 \log_2 N$:

   $$\log_2 N = \log_2 \left(2^{2^{16}}\right) = 2^{16}$$

   $$\log_2 \log_2 N = \log_2 \left(2^{16}\right) = 16$$

Thus, the answer is 16.