A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$.
1. **Identify the given conditions and translate them into mathematical expressions:**
   - The circle intersects the $y$-axis at points $(0, a)$ and $(0, b)$.
   - The circle is tangent to the line $x + 100y = 100$ at the point $(100, 0)$.

2. **Determine the equation of the tangent line:**
   - The line $x + 100y = 100$ can be rewritten in slope-intercept form as $y = -\frac{1}{100}x + 1$.
   - The slope of this line is $-\frac{1}{100}$.

3. **Find the center and radius of the circle:**
   - Since the circle is tangent to the line at $(100, 0)$, the radius of the circle is the perpendicular distance from the center of the circle to the line $x + 100y = 100$.
   - Let the center of the circle be $(h, k)$ and the radius be $r$.
   - The distance from $(h, k)$ to the line $x + 100y = 100$ is given by:
     \[
     \frac{|h + 100k - 100|}{\sqrt{1^2 + 100^2}} = r
     \]
     Simplifying, we get:
     \[
     \frac{|h + 100k - 100|}{\sqrt{10001}} = r
     \]

4. **Use the fact that the circle intersects the $y$-axis at $(0, a)$ and $(0, b)$:**
   - The equation of the circle can be written as:
     \[
     (x - h)^2 + (y - k)^2 = r^2
     \]
   - Substituting $x = 0$ into the circle's equation, we get:
     \[
     h^2 + (a - k)^2 = r^2 \quad \text{and} \quad h^2 + (b - k)^2 = r^2
     \]

5. **Solve for $a$ and $b$:**
   - Since the circle intersects the $y$-axis at $(0, a)$ and $(0, b)$, we have:
     \[
     a = k + \sqrt{r^2 - h^2} \quad \text{and} \quad b = k - \sqrt{r^2 - h^2}
     \]

6. **Calculate $ab - a - b$:**
   - First, find $ab$:
     \[
     ab = (k + \sqrt{r^2 - h^2})(k - \sqrt{r^2 - h^2}) = k^2 - (r^2 - h^2) = h^2 + k^2 - r^2
     \]
   - Next, find $a + b$:
     \[
     a + b = (k + \sqrt{r^2 - h^2}) + (k - \sqrt{r^2 - h^2}) = 2k
     \]
   - Therefore:
     \[
     ab - a - b = (h^2 + k^2 - r^2) - 2k
     \]

7. **Use the tangency condition to find $h$ and $k$:**
   - Since the circle is tangent to the line $x + 100y = 100$ at $(100, 0)$, we have:
     \[
     h = 100 \quad \text{and} \quad k = 0
     \]
   - The radius $r$ is the distance from $(100, 0)$ to the line $x + 100y = 100$:
     \[
     r = \frac{|100 + 100 \cdot 0 - 100|}{\sqrt{1^2 + 100^2}} = \frac{0}{\sqrt{10001}} = 0
     \]

8. **Substitute $h$, $k$, and $r$ into the expression for $ab - a - b$:**
   - Since $r = 0$, we have:
     \[
     ab - a - b = (100^2 + 0^2 - 0^2) - 2 \cdot 0 = 10000
     \]

The final answer is $\boxed{10000}$.