The rational choice depends on your utility function. Your utility function is unlikely to be linear with money. For example, if your utility function is log (X), then you will accept the first bet, be indifferent to the second bet, and reject the third bet. Any risk-averse utility function (i.e. any monotonically increasing function with negative second derivative) reaches a point where the agent stops playing the game.
A VNM-rational agent with a linear utility function over money will indeed always take this bet. From this, we can infer that linear utility functions do not represent the utility of humans.
(EDIT: The comments by Satt and AlexMennen are both correct, and I thank them for the corrections. I note that they do not affect the main point, which is that rational agents with standard utility functions over money will eventually stop playing this game)
For example, if your utility function is log (X), then you will accept the first bet
Not even that. You start with $1 (utility = 0) and can choose between
walking away with $1 (utility = 0), and
accepting a lottery with a 50% chance of leaving you with $0 (utility = −∞) and a 50% chance of having $3 (utility = log(3)).
The first bet's expected utility is then −∞, and you walk away with the $1.
You are at a casino. You have $1. A table offers you a game: you have to bet all your money; a fair coin will be tossed; if it lands heads, you triple your money; if it lands tails, you lose everything.
In the first round, it is rational to take the bet since the expected value of winning is $1.50, which is greater than what you started out with.
If you win the first round, you'll have $3. In the next round, it is rational to take the bet again, since the expected value is $4.50 which is larger than $3.
If you win the second round, you'll have $9. In the next round, it is rational to take the bet again, since the expected value is $13.50 which is larger than $9.
You get the idea. At every round, if you won the previous round, it is rational to take the next bet.
But if you follow this strategy, it is guaranteed that you will eventually lose everything. You will go home with nothing. And that seems irrational.
Intuitively, it feels that the rational thing to do is to quit while you are ahead, but how do you get that prediction out of the maximization of expected utility? Or does the above analysis only feel irrational because humans are loss-averse? Or is loss-aversion somehow optimal here?
Anyway, please dissolve my confusion.