Let's analyze the clues step by step to find this nine-digit number:

Clues:

1. Nine-digit number: The number has exactly nine digits.
2. Hundred-million digit is 3, and unit digit is 8: The number takes the form $3XXXXXXXX8$, where the first digit is 3, and the last digit is 8.
3. Rounding property: No matter how you round the number (to the nearest hundred million, ten million, etc.), you get the same rounded value.

Interpretation of the rounding property:

For the number to have the same rounded value regardless of the rounding level, it means that all digits except the hundred-million place should be as small as possible. Specifically, this would imply that all digits from the ten-million place to the ten place should be zero.

Reasoning:

• If rounding to the nearest hundred million: The number would remain 300,000,000.
• If rounding to the nearest ten million: The number would still be 300,000,000 (if the digits after the hundred-million place are 0).
• If rounding to the nearest million: Same result for the same reason, etc.

Thus, the number must be $300,000,008$.

Final Answer:

The nine-digit number is 300,000,008.

---

Would you like more details or have any further questions?

Here are five related questions to expand on this problem:

1. How does rounding work at different place values?
2. What are the properties of numbers that stay constant when rounded?
3. What are other examples of numbers with such rounding characteristics?
4. How can we generalize this approach to larger numbers?
5. Can the number change if the hundred-million digit is something else?

Tip: When rounding a number, always check the place value right after the place you're rounding to, as it determines whether to round up or down.