1. Understanding the Problem:
- We have a strip of width 1.
- The strip is divided into rectangular panels of width 1 and lengths $a_1, a_2, a_3, \ldots$.
- The panels are laid without gaps.
- From the second panel onwards, each panel is similar but not congruent to the already paved part of the strip.
- After laying the first $n$ slabs, the length of the paved part of the strip is $s_n$.
- We need to determine if there is a number that is not surpassed by any $s_n$.

2. Analyzing the Panels:
- The first panel has length $a_1$.
- The second panel is similar but not congruent to the first panel, meaning it is a scaled version of the first panel.
- Let the scaling factor be $k$. Then the length of the second panel is $k \cdot a_1$.
- The third panel is similar to the first two panels, so its length is $k^2 \cdot a_1$, and so on.

3. Generalizing the Lengths:
- The length of the $n$-th panel is $k^{n-1} \cdot a_1$.
- The total length of the paved part of the strip after $n$ panels is:
$$s_n = a_1 + k \cdot a_1 + k^2 \cdot a_1 + \cdots + k^{n-1} \cdot a_1$$

4. Summing the Series:
- The series $a_1 + k \cdot a_1 + k^2 \cdot a_1 + \cdots + k^{n-1} \cdot a_1$ is a geometric series with the first term $a_1$ and common ratio $k$.
- The sum of the first $n$ terms of a geometric series is given by:
$$s_n = a_1 \frac{1 - k^n}{1 - k} \quad \text{for} \quad k \neq 1$$

5. Behavior of the Series:
- If $0 < k < 1$, as $n$ approaches infinity, $k^n$ approaches 0.
- Therefore, the sum $s_n$ approaches:
$$s_n \to a_1 \frac{1}{1 - k}$$
- This means that there is a limit to the length of the paved part of the strip, which is $a_1 \frac{1}{1 - k}$.

6. Conclusion:
- Given $a_1$, there is indeed a number that is not surpassed by any $s_n$, which is $a_1 \frac{1}{1 - k}$.

The final answer is $\boxed{ a_1 \frac{1}{1 - k} }$.