7. In the polar coordinate system, the distance from the pole to the center of the circle $\rho=3 \sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)+7 \sin \theta$ is
7. $\frac{5}{2}$

Since $\rho=3 \cos \theta+4 \sin \theta$, we have $\rho^{2}=3 \rho \cos \theta+4 \rho \sin \theta$.
Therefore, the equation of the circle in Cartesian coordinates is $x^{2}+y^{2}=3 x+4 y$, so the distance from the origin (i.e., the pole) to the center of the circle $\left(-\frac{3}{2},-2\right)$ is $\sqrt{2^{2}+\left(\frac{3}{2}\right)^{2}}=\frac{5}{2}$.