5. As shown in Figure 1, given

Ellipse $C: \frac{x^{2}}{25}+y^{2}=1$ and $\odot O: x^{2}+y^{2}=1$, the maximum radius of a circle contained within the region inside ellipse $C$ and outside $\odot O$ (including the boundaries) is $\qquad$
5. $\frac{23}{25}$.

To maximize the radius $r$ of the circle, whose center lies on the $x$-axis and is externally tangent to $\odot O$, and intersects the ellipse $C$ at only one point in the first quadrant.
By symmetry, let the equation of the desired circle $M$ be
$$
\begin{array}{l}
(x-r-1)^{2}+y^{2}=r^{2} \\
\Rightarrow y^{2}=r^{2}-(x-r-1)^{2} .
\end{array}
$$

Substituting into the ellipse equation yields
$$
\begin{array}{l}
\frac{x^{2}}{25}+r^{2}-(x-r-1)^{2}=1 \\
\Rightarrow 24 x^{2}-50(r+1) x+50(r+1)=0 . \\
\text { By } \Delta=2500(r+1)^{2}-4800(r+1)=0 \\
\Rightarrow r=\frac{23}{25} .
\end{array}
$$