Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
1. **Understanding the Problem:**
   We need to determine the maximal number of \(120^\circ\) angles in a 7-gon inscribed in a circle, where all sides are of different lengths.

2. **Initial Claim:**
   We claim that the maximum number of \(120^\circ\) angles is 2.

3. **Setup:**
   Consider a cyclic heptagon \(ABCDEFG\). If two adjacent angles are equal, then the sides are symmetric with respect to their perpendicular bisector, contradicting our original condition of different-length sides.

4. **Case Analysis:**
   - Suppose there are 3 non-adjacent angles equal to \(120^\circ\), say \(\angle B = \angle D = \angle F = 120^\circ\).
   - Let \(O\) be the circumcenter of the circle.

5. **Angle Calculation:**
   - Since \(\angle B = 120^\circ\), the arc subtended by \(\angle B\) is \(240^\circ\) (since the central angle is twice the inscribed angle).
   - Similarly, \(\angle D = 120^\circ\) implies the arc subtended by \(\angle D\) is \(240^\circ\).
   - \(\angle F = 120^\circ\) implies the arc subtended by \(\angle F\) is \(240^\circ\).

6. **Contradiction:**
   - If \(\angle B = \angle D = \angle F = 120^\circ\), then the central angles \(\angle COA = \angle COE = \angle EOG = 120^\circ\).
   - This implies that the sum of these central angles is \(360^\circ\), which is the total angle around point \(O\).
   - However, this configuration would force \(\angle GOA = 0^\circ\), which is impossible in a geometric figure.

7. **Conclusion:**
   - Therefore, having 3 non-adjacent \(120^\circ\) angles leads to a contradiction.
   - Hence, the maximum number of \(120^\circ\) angles in such a 7-gon is 2.

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