5. In the Cartesian coordinate system, let $O$ be the origin, point $A(-1,0), B(0, \sqrt{3})$, and the moving point $C$ lies on the circle
$$
(x-3)^{2}+y^{2}=4
$$

Then the maximum value of $|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}|$ is $\qquad$.
5. $\sqrt{7}+2$.
Let $C(3+2 \cos \theta, 2 \sin \theta)(\theta \in[0,2 \pi))$. Then
$$
\begin{array}{l}
|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}| \\
=\sqrt{(2+2 \cos \theta)^{2}+(\sqrt{3}+2 \sin \theta)^{2}} \\
\leqslant \sqrt{2^{2}+(\sqrt{3})^{2}}+\sqrt{(2 \cos \theta)^{2}+(2 \sin \theta)^{2}} \\
=\sqrt{7}+2 .
\end{array}
$$

The equality holds if and only if the line connecting the point $(\sqrt{3}, 2)$ and $(2 \cos \theta, 2 \sin \theta)$ passes through the origin $O$. This is clearly achievable.