Determine the least integer $k$ for which the following story could hold true:
In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
To determine the least integer \( k \) for which the given conditions hold true, we need to analyze the constraints and prove the feasibility of the solution.

1. **Analyzing the constraints for \( k = 3 \):**
   - Each player plays between 2 and 3 games against each of the other 23 players.
   - Therefore, the total number of games played by each player ranges from \( 2 \times 23 = 46 \) to \( 3 \times 23 = 69 \).
   - Since there are 24 players, each must have played a different number of games. This means we need 24 distinct integers in the range from 46 to 69.
   - However, the range from 46 to 69 contains only 24 numbers: \( 46, 47, \ldots, 69 \).
   - If one player plays 46 games (2 games against each of the other 23 players), and another plays 69 games (3 games against each of the other 23 players), it is impossible for both conditions to be satisfied simultaneously. This is because the player with 69 games would have played 3 games against the player with 46 games, which contradicts the latter's total of 46 games.
   - Hence, \( k = 3 \) is not possible.

2. **Proving \( k = 4 \) is possible by induction:**
   - **Base Case (\( n = 3 \)):**
     - Consider three players who play 2, 3, and 4 games against each other.
     - The total number of games played by these players will be 5, 6, and 7 respectively.
     - This satisfies the condition that each player plays a different number of games.

   - **Inductive Step:**
     - Assume that for \( n \geq 3 \), there exists a tournament \( T_n \) with \( n \) players where each pair of players plays between 2 and 4 games, and each player plays a different number of games.
     - Let the number of games played by the players be \( a_1 < a_2 < \cdots < a_n \).

     - **Case (i):** If no player has played exactly 2 games against every other player:
       - This implies \( a_1 > 2n - 2 \).
       - Introduce a new player who plays exactly 2 games against each of the \( n \) existing players.
       - The new number of games played by this player is \( 2n \).
       - The new sequence of games played will be \( 2n, a_1 + 2, a_2 + 2, \ldots, a_n + 2 \).
       - This sequence maintains the property that each player plays a different number of games.

     - **Case (ii):** If no player has played exactly 4 games against every other player:
       - This implies \( a_n < 4n - 4 \).
       - Introduce a new player who plays exactly 4 games against each of the \( n \) existing players.
       - The new number of games played by this player is \( 4n \).
       - The new sequence of games played will be \( a_1 + 4, a_2 + 4, \ldots, a_n + 4, 4n \).
       - This sequence also maintains the property that each player plays a different number of games.

   - By induction, we have shown that for every \( n \geq 3 \), there exists a tournament \( T_n \) with \( n \) players where each pair of players plays between 2 and 4 games, and each player plays a different number of games.

\(\blacksquare\)

The final answer is \( \boxed{ k = 4 } \).