4. If $x \in[0,1]$, and $\log _{2} \log _{2}(2 x+2)+2^{2 x+2}$ is an integer, then the number of real numbers $x$ that satisfy this condition is
A. 12
B. 13
C. 14
D. 15
$$
\begin{array}{l}
2 x+2 \in[2,4] \Rightarrow \log _{2}(2 x+2) \in[1,2] \Rightarrow \log _{2} \log _{2}(2 x+2) \in[0,1], 2^{2 x+2} \in[4,16] \\
\Rightarrow \log _{2} \log _{2}(2 x+2)+2^{2 x+2} \in[4,17], \text { which includes } 14 \text { integers. Therefore, the answer is } C .
\end{array}
$$