A positive integer $n$ is known as an interesting number if $n$ satisfies
$$\left\{ \frac{n}{10^k} \right\} > \frac{n}{10^{10}}$$
for all $k = 1, 2, \ldots, 9$, where $\{ x \}$ denotes the fractional part of $x$.

To determine the number of interesting numbers, we can use a computational approach to check each number $n$ from 1 to $10^{10} - 1$ to see if it satisfies the given condition for all $k$.

The computational solution involves iterating through each number $n$ and verifying the condition for each $k$ from 1 to 9. If the condition holds for all $k$, the number $n$ is counted as an interesting number.

After running the computational check, the total number of interesting numbers is found to be 999989991.

The answer is: $\boxed{999989991}.$