In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played?
1. In this tournament, each player must be defeated three times to be eliminated. Therefore, each of the 511 players who do not win will be defeated exactly three times.
2. The winner of the tournament will be defeated at most 2 times, as they are not eliminated.
3. To find the total number of games played, we calculate the total number of defeats:
   \[
   \text{Total defeats} = (511 \text{ players} \times 3 \text{ defeats per player}) + 2 \text{ defeats for the winner}
   \]
4. Performing the multiplication and addition:
   \[
   511 \times 3 = 1533
   \]
   \[
   1533 + 2 = 1535
   \]

Conclusion:
The greatest number of games that could be played is \(\boxed{1535}\).