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Once the player loses, the chain ends, and the house wins. So as long as the house can afford to keep pushing the player in to trying again, they're going to create more opportunities for the player to return their winnings to the house.
Right, and as the chain continues, the probability of the player maintaining their streak becomes infinitesimal. But the potential payout scales at the same rate.
If the player goes for 3 rounds, they only have a 1/8 chance of winning... but they'll get 8 times their initial bet. So it's technically a fair game, right?
They have a 50/50 chance each round. Doesn't matter how many rounds they've won.
I'm looking at the game as a whole. The player has a 1 in 8 chance of winning 3 rounds overall.