[jason1990]

Another way to get very close to the true expected value is to reason as follows: Define a "round" of the game to be a sequence of flips that either ends with getting a point or ends with flipping a tail. So if your first flip is a tail, then the first round is over and the next round begins. But if your first flip is a head and the second flip is a tail, then the first round is over after the second flip and the next round begins. If your first two flips are heads, then the first round will be over after the third flip, no matter what happens.

It's easy to see that the expected length of each round is 1.75 flips. So you can expect to play 10000/1.75 rounds. In each round, you get a point with probability 1/8, so you expect to get (10000/1.75)*(1/8) = 10000/14 points.

Notice that this is not exactly the right answer, but it differs from the true expected value by less than than 1/50th of a percent. The discrepancy comes from the fact that you may not be able to complete the final round, since you are not allowed any more than 10000 flips.