The Grinder's Bankroll

A casino hands you a bankroll of 1000 chips and one house rule: before you're allowed to walk away, every one of those 1000 chips must have been put at risk in at least one wager. You choose your stake size $x$ freely on each round. Every wager is settled by a fair coin flip — win and you keep your stake plus a payout of 90% of it; lose and the stake is gone. How should you size your bets to walk out with as many chips as possible, in expectation?