1. Identify the problem type: The problem is about finding the largest unattainable amount using two given denominations of stamps. This is a classic problem that can be solved using the Chicken McNugget Theorem (also known as the Frobenius Coin Problem).

2. State the Chicken McNugget Theorem: The theorem states that for two coprime integers $a$ and $b$, the largest integer that cannot be expressed as a non-negative integer combination of $a$ and $b$ is given by:
$$n = ab - a - b$$
Here, $a = 8$ and $b = 15$.

3. Verify that the numbers are coprime: We need to check that 8 and 15 are coprime, i.e., their greatest common divisor (gcd) is 1.
$$\gcd(8, 15) = 1$$
Since 8 and 15 are coprime, we can apply the Chicken McNugget Theorem.

4. Apply the theorem: Using the theorem, we calculate the largest unattainable amount:
$$n = 8 \cdot 15 - 8 - 15$$
$$n = 120 - 8 - 15$$
$$n = 120 - 23$$
$$n = 97$$

5. Conclusion: The largest unattainable amount of postage using 8-cent and 15-cent stamps is 97 cents.

The final answer is $\boxed{97}$