We are given a positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2$, where $s(n)$ denotes the sum of the digits of $n$. We need to find the largest positive integer $n$ that satisfies this equation.

1. Define the equation and constraints:
$$n = S^2 + 2S - 2$$
where $S = s(n)$ is the sum of the digits of $n$.

2. **Determine the number of digits $k$ in $n$:**
Since $n$ has $k$ digits, we have:
$$10^{k-1} \leq n < 10^k$$
Also, the sum of the digits $S$ satisfies:
$$S \leq 9k$$

3. **Substitute $n$ in the inequality:**
$$10^{k-1} \leq S^2 + 2S - 2$$
and
$$S \leq 9k$$

4. **Estimate the upper bound for $k$:**
$$10^{k-1} < (9k + 1)^2$$
We need to find the largest $k$ such that this inequality holds. Testing values of $k$:

- For $k = 3$:
$$10^2 = 100 < (9 \cdot 3 + 1)^2 = 28^2 = 784$$
This holds true.

- For $k = 4$:
$$10^3 = 1000 < (9 \cdot 4 + 1)^2 = 37^2 = 1369$$
This also holds true, but we need to check if $k = 4$ provides a valid solution.

5. Check the divisibility condition:
Since $n \equiv s(n) \pmod{9}$, we have:
$$n - S \equiv 0 \pmod{9}$$
Substituting $n = S^2 + 2S - 2$:
$$S^2 + 2S - 2 - S \equiv 0 \pmod{9}$$
$$S^2 + S - 2 \equiv 0 \pmod{9}$$
Solving $S^2 + S - 2 \equiv 0 \pmod{9}$:
$$S(S + 1) \equiv 2 \pmod{9}$$
Testing values of $S$ modulo 9, we find:
$$S \equiv 1, 7 \pmod{9}$$

6. Combine constraints:
Since $10 \leq S \leq 27$ and $S \equiv 1, 7 \pmod{9}$, possible values for $S$ are:
$$S \in \{10, 16, 19, 25\}$$

7. **Calculate $n$ for each $S$:**
$$n = S^2 + 2S - 2$$
- For $S = 10$:
$$n = 10^2 + 2 \cdot 10 - 2 = 100 + 20 - 2 = 118$$
- For $S = 16$:
$$n = 16^2 + 2 \cdot 16 - 2 = 256 + 32 - 2 = 286$$
- For $S = 19$:
$$n = 19^2 + 2 \cdot 19 - 2 = 361 + 38 - 2 = 397$$
- For $S = 25$:
$$n = 25^2 + 2 \cdot 25 - 2 = 625 + 50 - 2 = 673$$

8. Verify the sum of digits:
- For $n = 118$, $s(118) = 1 + 1 + 8 = 10$
- For $n = 286$, $s(286) = 2 + 8 + 6 = 16$
- For $n = 397$, $s(397) = 3 + 9 + 7 = 19$
- For $n = 673$, $s(673) = 6 + 7 + 3 = 16$ (does not match)

Therefore, the valid solutions are $n = 118, 286, 397$.

Conclusion:
The largest natural number $n$ which satisfies the equation is $n = 397$.

The final answer is $\boxed{397}$