Coin Game

You're invited to play a coin toss game with unique rules. The game uses a fair coin, and each player scores points based on the coin's outcome: if it lands tails up, you earn a point; if it lands heads up, your opponent scores a point. The game concludes when a player has achieved a two-point lead over their competitor, upon which the losing party must pay the winner one euro. A twist in the game allows you, at any time before the match ends, to invoke an option to double the final payout. However, this can only be exercised once. Given these rules and the ability to strategically double the payout, what is the maximum amount you would be willing to pay for the privilege of participating in this game?

Answer format: round to 3 decimal places.