3. $\frac{\pi+2}{\pi-2}$.

Let's assume the center of the circle falls in $Z$ as shown in Figure 7(a).
When the chord $AB$ moves upward, the shaded area in Figure 7(b) is greater than the unshaded area to its left, so $S_{X}+S_{Z}$ increases, while $S_{Y}+S_{W}$ decreases (note that the sum of the areas of $X$, $Y$, $Z$, and $W$ is a constant $\left.\pi r^{2}\right)$. Therefore, the ratio $\frac{S_{X}+S_{Z}}{S_{Y}+S_{W}}$ increases.

Thus, when point $A$ coincides with point $C$, it is possible to achieve the maximum value.

In Figure 7(c), in the right triangle $\triangle ABD$, the hypotenuse $BD$ is the diameter, so $\triangle ABD$ has the maximum area when $OA$ is the height. At this point, $S_{Z}$ is maximized, and $S_{X}+S_{Z}$ is also maximized, with a value of $\frac{1}{2} \pi r^{2}+r^{2}$. Meanwhile, $S_{Y}+S_{W}$ is minimized, with a value of $\frac{1}{2} \pi r^{2}-r^{2}$.
Therefore, the maximum value of $\frac{S_{X}+S_{Z}}{S_{Y}+S_{W}}$ is
$$\frac{\frac{1}{2} \pi r^{2}+r^{2}}{\frac{1}{2} \pi r^{2}-r^{2}}=\frac{\pi+2}{\pi-2} .$$