  We axiomatize the first-order theories of exponential integer parts of
real-closed exponential fields in a language with $2^x$, in a language with a
predicate for powers of 2, and in the basic language of ordered rings. In
particular, the last theory extends IOpen by sentences expressing the existence
of winning strategies in a certain game on integers; we show that it is a
proper extension of IOpen, and give upper and lower bounds on the required
number of rounds needed to win the game.
