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 the condition is ( ).
(A) 12
(B) 13
(C) 14
(D) 15
4. C.
$$
\begin{array}{l}
\text { Let } f(x) \\
=\log _{2} \log _{2}(2 x+2)+2^{2 x+2}(x \in[0,1]) \text {. }
\end{array}
$$

Then $f(x)$ is an increasing function, $f(0)=4, f(1)=17$.
Therefore, the range of $f(x)$ is $[4,17]$.
Thus, $f(x)$ takes 14 integer values, and there are 14 corresponding $x$ values.