For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

1. **Define the problem and variables:**
   - Let \( n \ge 3 \) be the number of sides of the polygon.
   - We draw \( n - 3 \) internal diagonals, dividing the \( n \)-gon into \( n - 2 \) triangles.
   - Let \( k = n - 2 \) be the number of triangles formed.
   - Each angle in these triangles is a natural number, and no two angles are equal.

2. **Calculate the total sum of the internal angles:**
   - Each triangle has an angle sum of \( 180^\circ \).
   - Therefore, the total sum of the angles in \( k \) triangles is \( 180^\circ k \).

3. **Sum of distinct natural number angles:**
   - The angles are distinct natural numbers from \( 1^\circ \) to \( 3k^\circ \).
   - The sum of the first \( 3k \) natural numbers is given by the formula for the sum of an arithmetic series:
     \[
     \sum_{i=1}^{3k} i = \frac{3k(3k + 1)}{2}
     \]

4. **Set up the inequality:**
   - Since the total sum of the angles in the triangles must be at least the sum of the first \( 3k \) natural numbers:
     \[
     180k \ge \frac{3k(3k + 1)}{2}
     \]

5. **Simplify the inequality:**
   \[
   180k \ge \frac{3k(3k + 1)}{2}
   \]
   \[
   360k \ge 3k(3k + 1)
   \]
   \[
   360k \ge 9k^2 + 3k
   \]
   \[
   0 \ge 9k^2 - 357k
   \]
   \[
   0 \ge k(9k - 357)
   \]
   \[
   9k - 357 \le 0
   \]
   \[
   k \le \frac{357}{9}
   \]
   \[
   k \le 39.6667
   \]
   Since \( k \) must be an integer, the largest possible value for \( k \) is \( 39 \).

6. **Determine \( n \):**
   \[
   k = n - 2
   \]
   \[
   39 = n - 2
   \]
   \[
   n = 41
   \]

7. **Verify the construction:**
   - We need to ensure that we can construct \( 41 \)-gon with the given properties.
   - A possible construction is gluing together triangles with internal angles \( i^\circ, (i+59)^\circ, (121-2i)^\circ \) for \( 1 \le i \le 20 \), and \( j^\circ, (j+20)^\circ, (160-2j)^\circ \) for \( 21 \le j \le 39 \).
   - In the first set, \( i < i+59 < 121-2i \).
   - In the second set, \( j < j+20 < 160-2j \).
   - For triangles from two different sets, \( i < j < j+20 \le 59 < i+59 \le 79 < 82 \le 160-2j \), and \( 121-2i \) and \( 160-2j \) have different parity.

The final answer is \( \boxed{41} \).