$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw  $0.5$ points, for defeat  $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?
1. **Calculate the total number of games played:**
   Since there are 16 players and each pair of players plays exactly one game, the total number of games played is given by the combination formula:
   \[
   \binom{16}{2} = \frac{16 \times 15}{2} = 120
   \]
   Each game results in a total of 1 point being distributed (either 1 point to the winner or 0.5 points to each player in case of a draw).

2. **Calculate the total number of points distributed:**
   Since there are 120 games and each game contributes 1 point to the total score, the total number of points distributed among all players is:
   \[
   120 \text{ points}
   \]

3. **Determine the average score per player:**
   The average score per player is:
   \[
   \frac{120 \text{ points}}{16 \text{ players}} = 7.5 \text{ points per player}
   \]

4. **Analyze the scores of the 15 players who shared the first place:**
   Since 15 players shared the first place, their scores must be strictly higher than the average score of 7.5 points. The maximum possible score for any player is 15 points (if they won all their games), but since they all shared the first place, their scores must be equal and higher than 7.5 points.

5. **Determine the possible scores for the 15 players:**
   The only integer score higher than 7.5 and less than or equal to the maximum possible score is 8 points. Therefore, each of the 15 players must have scored 8 points.

6. **Calculate the total points scored by the 15 players:**
   \[
   15 \text{ players} \times 8 \text{ points per player} = 120 \text{ points}
   \]

7. **Determine the score of the sixteenth player:**
   Since the total number of points distributed is 120 and the 15 players together scored 120 points, the sixteenth player must have scored:
   \[
   120 \text{ points} - 120 \text{ points} = 0 \text{ points}
   \]

Conclusion:
The sixteenth player scored 0 points.

The final answer is $\boxed{0}$.