Let $ABCD$ be a quadrilateral with side lengths $AB = 2$, $BC = 3$, $CD = 5$, and $DA = 4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$?
1. **Identify the problem**: We need to find the maximum possible radius of a circle inscribed in a quadrilateral \(ABCD\) with given side lengths \(AB = 2\), \(BC = 3\), \(CD = 5\), and \(DA = 4\).

2. **Determine the type of quadrilateral**: The area of a quadrilateral is maximized when it is cyclic. A cyclic quadrilateral is one where all vertices lie on a single circle.

3. **Use Brahmagupta's formula**: For a cyclic quadrilateral with sides \(a\), \(b\), \(c\), and \(d\), the area \(K\) is given by:
   \[
   K = \sqrt{(s-a)(s-b)(s-c)(s-d)}
   \]
   where \(s\) is the semiperimeter:
   \[
   s = \frac{a + b + c + d}{2}
   \]

4. **Calculate the semiperimeter**:
   \[
   s = \frac{2 + 3 + 5 + 4}{2} = 7
   \]

5. **Substitute the side lengths into Brahmagupta's formula**:
   \[
   K = \sqrt{(7-2)(7-3)(7-5)(7-4)} = \sqrt{5 \cdot 4 \cdot 2 \cdot 3} = \sqrt{120} = 2\sqrt{30}
   \]

6. **Find the inradius**: The radius \(r\) of the inscribed circle in a cyclic quadrilateral is given by:
   \[
   r = \frac{K}{s}
   \]
   Substituting the values we have:
   \[
   r = \frac{2\sqrt{30}}{7}
   \]

Conclusion:
\[
\boxed{\frac{2\sqrt{30}}{7}}
\]