Game 1 gives you 200 points every time you
score. Game 2 doubles your points every time you
score (2, 4, 8, and so on). In which game do you
think you will you earn more points?

its depends on how many times you score during the game 

in second game the number of points increases  as a geometric sequence

with common ratio 2

so for example if you score ten times in first game you get 2000 points

if you score 10 times in Game 2 you score   2 * (2^10) - 1  =  2046 points

so playing 10 or more games  is best with Game 2.  Any less plays favours gane 1.