Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?
1. **Understanding the Problem:**
   We need to find the greatest possible product of the digits of a positive integer \( N \) whose digits sum up to 23.

2. **Strategy:**
   To maximize the product of the digits, we should use the digits in such a way that their product is maximized. Generally, the digit 9 is the most efficient in terms of maximizing the product, but we need to ensure the sum of the digits is 23.

3. **Using Digits Efficiently:**
   - The digit 9 is the largest single-digit number and will contribute significantly to the product.
   - We need to find a combination of digits that sum to 23 and maximize the product.

4. **Initial Calculation:**
   - If we use two 9's, their sum is \( 9 + 9 = 18 \).
   - We need additional digits that sum to \( 23 - 18 = 5 \).

5. **Finding the Remaining Digits:**
   - The remaining sum is 5. The digits that sum to 5 and maximize the product are 5 itself (since 5 is greater than any combination of smaller digits that sum to 5).

6. **Combining the Digits:**
   - Using two 9's and one 5, the digits are \( 9, 9, \) and \( 5 \).
   - The product of these digits is \( 9 \times 9 \times 5 = 81 \times 5 = 405 \).

7. **Verification:**
   - The sum of the digits \( 9 + 9 + 5 = 23 \).
   - The product is \( 405 \).

8. **Exploring Other Combinations:**
   - If we use one 9, the remaining sum is \( 23 - 9 = 14 \).
   - The digits that sum to 14 and maximize the product are \( 8 + 6 \) (since \( 8 \times 6 = 48 \) is greater than any other combination of digits summing to 14).

9. **Combining the Digits:**
   - Using one 9, one 8, and one 6, the digits are \( 9, 8, \) and \( 6 \).
   - The product of these digits is \( 9 \times 8 \times 6 = 72 \times 6 = 432 \).

10. **Verification:**
    - The sum of the digits \( 9 + 8 + 6 = 23 \).
    - The product is \( 432 \).

11. **Conclusion:**
    - The combination \( 9, 8, 6 \) provides a higher product than \( 9, 9, 5 \).

The final answer is \(\boxed{432}\)