Doubling Strategy and Optional Stopping Failure

A gambler starts with $0 net and makes fair$1-doubling bets (bet 1, then 2, then 4, ...) on a sequence of fair coin flips, stopping the first time they win a single flip (guaranteeing +$1 net). Let T be that stopping time. Compute E[net wealth at T], and explain whether E[net at T] equals net at time 0 as naive optional stopping would suggest.