In a Women's Chess Championship having 300 entrants, a player is eliminated whenever she loses a match. It is given that no match results in a draw. The number of matches that are played in the entire Championship is:

As there are total 300 players in the tournament and the goal is to eliminate 299 players to announce the final winner . 
It is given that every single match will eliminate one player as there are no draws . So, in order to eliminate 299 players in the tournament, we need 299 matches to be conducted .

Alternate Solution : 
Round 1 : We can conduct 150 matches among 300 players each playing exactly once , so that we can eliminate all the 150 players who lost their respective matches .
                                       Remaining players in tournament = 150

Round 2 : Now we can conduct 75 matches among 150 players each playing exactly once, so that we can eliminate 75 players who lost their matches.
                            Remaining players = 75

Round 3 : Now we can conduct 37 matches for 74 people and make 1 player sit idle without any match. In this round we can eliminate 37 players.
                            Remaining players = 37+1 = 38.

Round 4 : Now we can conduct 19 matches for these 38 players , and eliminate 19 players who loose their matches.
                            Remaining players = 19.

Round 5 : Now we can conduct 9 matches for 18 players and make 1 player sit idle in this round. So we can eliminate 9 players here.
                            Remaining players = 9+1 = 10.

Round 6 : Now we can conduct 5 matches for 10 players and eliminate 5 players .
                            Remaining players = 5 .

Round 7 : we can conduct 2 matches and make 1 player sit idle in this round. So we can eliminate 2 players .
                            Remaining players = 2 + 1 = 3.

Round 8 : We can conduct 1 match to eliminate one player .
                            Remaining players = 2.

Round 9 : We can conduct last 1 match to eliminate one among 2.

Therefore, total matches = 150 + 75 + 37 + 19 + 9 + 5 + 2 + 1 + 1 = 299