Solution 1:

WITH CTE(player_id, event_date,games_played) AS
(
  SELECT  1 , '2016-03-01' , 5 UNION ALL
  SELECT  1 , '2016-05-02' , 6 UNION ALL
  SELECT 1 , '2017-06-25' , 1 UNION ALL
  SELECT 3 , '2016-03-02' , 12 UNION ALL
  SELECT 3 , '2018-07-03' , 5  
)
SELECT C.player_id,C.event_date,C.games_played,
 SUM(C.games_played)OVER(PARTITION BY C.player_id ORDER BY C.Event_date ASC)games_played_so_far
FROM CTE AS C
ORDER BY C.player_id

If this solution is not applicable to snowflake then please google something like "running total for snowflake"