Square $S_1$ is inscribed inside circle $C_1$, which is inscribed inside square $S_2$, which is inscribed inside circle $C_2$, which is inscribed inside square $S_3$, which is inscribed inside circle $C_3$, which is inscribed inside square $S_4$.

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Let $a$ be the side length of $S_4$, and let $b$ be the side length of $S_1$. What is $\tfrac{a}{b}$?
1. **Understanding the problem**: We have a sequence of squares and circles inscribed within each other. We need to find the ratio of the side length of the largest square \( S_4 \) to the smallest square \( S_1 \).

2. **Relationship between the squares and circles**:
   - For a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
   - For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square.

3. **Side length relationships**:
   - Let the side length of \( S_1 \) be \( b \).
   - The diagonal of \( S_1 \) is \( b\sqrt{2} \), which is also the diameter of \( C_1 \).
   - The side length of \( S_2 \) is equal to the diameter of \( C_1 \), so the side length of \( S_2 \) is \( b\sqrt{2} \).

4. **Continuing the pattern**:
   - The diagonal of \( S_2 \) is \( b\sqrt{2} \cdot \sqrt{2} = b \cdot 2 \), which is the diameter of \( C_2 \).
   - The side length of \( S_3 \) is equal to the diameter of \( C_2 \), so the side length of \( S_3 \) is \( b \cdot 2 \).

5. **Final step**:
   - The diagonal of \( S_3 \) is \( b \cdot 2 \cdot \sqrt{2} = b \cdot 2\sqrt{2} \), which is the diameter of \( C_3 \).
   - The side length of \( S_4 \) is equal to the diameter of \( C_3 \), so the side length of \( S_4 \) is \( b \cdot 2\sqrt{2} \).

6. **Finding the ratio**:
   - The ratio of the side length of \( S_4 \) to \( S_1 \) is:
     \[
     \frac{a}{b} = \frac{b \cdot 2\sqrt{2}}{b} = 2\sqrt{2}
     \]

The final answer is \(\boxed{2\sqrt{2}}\)