The "triple-or-nothing paradox" is that a game where I expect to increase my money (on average) at each stage ends up bankrupting me with probability 1 if I play long enough. However, the paradox seems to have an easy resolution, in the sense that a rational actor could decide that a 50% risk of losing your entire stake on each round is simply too high.
A slight modification removes the "easy way out" for this rational actor: I stake an initial wager W to play this game. At each coin flip, I either lose 75% of my current stake, or gain 125% of my current stake, with 50-50 probability. Any individual round of this game is quite attractive to a bettor, for the following reasons:
at every wager, the amount the bettor stands to win is bigger than the amount they stand to lose
the bettor never loses their entire stake (if we consider continuous quantities of money) and thus can always make a comeback
the bettor's expected value of their wager is 1/2 (125% - 75%) = 25% gain
expected value after n rounds is (1.25)^n of initial stake -- the bettor's expectation increases exponentially
At each round, the bettor has to choose between playing again and cashing out. The arguments above seem to favor playing again for any individual round. Yet over time, the bettor's wealth approaches 0 with probability 1: because it multiplies by 2.25 on wins, but divides by 4 on losses, more than 1 win is required to balance out a loss. Since the flips are fair, this means the bettor's wealth is multiplied on average by (2.25)/4 = 9/16 = 56.25% for every 2 rounds, which translates to a 3/4 = 75% multiplier per round on average. In other words, the bettor's actual wealth (as opposed to their expected wealth) multiplies by about 0.75 on each round, hence approaches 0 for large numbers of rounds.
The paradox is then that during the game, the bettor always has a reason to play the next round in terms of expectation. It is always irrational for them to stop playing after any particular round. Thus, a rational actor will decide to continue playing forever, whereupon their initial stake dwindles to zero with probability 1. How is it that making the most rational decision at each stage (whether one is currently winning or losing) leads to going broke?
Edit: Some commenters have attempted to resolve the paradox by appealing to an individual’s utility differing from the strict monetary payoffs. However, this attempted resolution utterly fails: if we know an individual’s utilities, we can always set up a version of this paradox that applies to them specifically. They “stake” something worth a certain amount of utility to them, and at each stage of the game, either lose 75% of the utility value of their current stake, or gain 125% of the utility value of their current stake. Once the game has been adjusted to their utility preferences, they face the same paradox as before. The presentation in terms of monetary value is for expositional purposes only and in no way fundamental to the paradox itself.