Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers. 

1. We start by considering the expression $\frac{2^a - 2^b}{2^c - 2^d}$, where $a, b, c, d$ are non-negative integers. Without loss of generality, we can assume $a > b$ and $c > d$.

2. For the expression to be an integer, the denominator $2^c - 2^d$ must divide the numerator $2^a - 2^b$. We can factor both the numerator and the denominator:
   \[
   2^a - 2^b = 2^b (2^{a-b} - 1)
   \]
   \[
   2^c - 2^d = 2^d (2^{c-d} - 1)
   \]

3. For $\frac{2^a - 2^b}{2^c - 2^d}$ to be an integer, $2^d (2^{c-d} - 1)$ must divide $2^b (2^{a-b} - 1)$. This implies that $2^d$ must divide $2^b$, which means $d \leq b$.

4. Additionally, $2^{c-d} - 1$ must divide $2^{a-b} - 1$. This is a consequence of the fact that the non-power-of-2 factors of the numerator and the denominator must match. Therefore, we need $c-d \mid a-b$.

5. Let $x = c - d$, $y = b - d$, and $z = \frac{a - b}{c - d}$. Then, the expression becomes:
   \[
   \frac{2^a - 2^b}{2^c - 2^d} = \frac{2^{d+y+zx} - 2^{d+y}}{2^d (2^x - 1)} = 2^d \cdot \frac{2^{y+zx} - 2^y}{2^x - 1}
   \]

6. Simplifying further, we get:
   \[
   \frac{2^{y+zx} - 2^y}{2^x - 1} = 2^y \cdot \frac{2^{zx} - 1}{2^x - 1}
   \]
   The term $\frac{2^{zx} - 1}{2^x - 1}$ is an integer because $2^x - 1$ divides $2^{zx} - 1$.

7. Therefore, the expression $\frac{2^a - 2^b}{2^c - 2^d}$ can be simplified to an integer if $c-d \mid a-b$ and $d \leq b$.

8. To find the smallest positive integer that cannot be expressed in this form, we need to check the smallest odd numbers, as even numbers can be expressed by multiplying an odd number by 2.

9. Checking the smallest odd numbers:
   - $1 = \frac{2^1 - 2^0}{2^1 - 2^0}$
   - $3 = \frac{2^2 - 2^0}{2^1 - 2^0}$
   - $5 = \frac{2^3 - 2^0}{2^2 - 2^0}$
   - $7 = \frac{2^3 - 2^0}{2^1 - 2^0}$
   - $9 = \frac{2^4 - 2^0}{2^2 - 2^0}$

10. However, $11$ cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$ for any non-negative integers $a, b, c, d$.

The final answer is $\boxed{11}$.