Since the coin flips after the first are made with a different coin, it doesn't matter whether that coin is fair. You probably can conclude that it's unfair, but as long as it's capable of coming down tails at all, the probability of surviving the game indefinitely is still 50%.

But biased coins do not exist short of the total bias of a double-headed coin. Biased coin-flippers do. So since it's you flipping your own coin, 1000 heads means you rigged the game.