I think the interesting question is what to do when you expect many more, but only finitely many rounds. It seems like Kelly should somehow gradually transition, until it recommends normal utility maximization in the case of only a single round happening ever. Log utility doesn't do this. I'm not sure I have anything that does though, so maybe it's unfair to ask it from you, but still it seems like a core part of the idea, that the Kelly strategy comes from the compounding, is lost.
Kelly betting seems somehow less justified when we're not doing it a bunch of times. If I were making bets left and right, I would feel more inclined to use Kelly; I could visualize the growth-maximizing behavior, and know that if I trusted my own probability assessments, I'd see that growth curve with high probability.
Several prediction markets have recently offered a bet at around 62¢ which superforecasters assign around 85% probability. This resulted in a rare temptation for me to Kelly bet. Calculating the Kelly formula, I found that I was supposed to put 60% of my bankroll on this.
Now, 60% of my savings seems like a lot. But am I really more risk-averse than Kelly? It struck me that if I were to do this sort of thing all the time, I would feel more strongly justified using Kelly.
Bunthut is suggesting a generalization of Kelly which accounts for the number of times we expect to iterate investment. In Bunthut's suggestion, we would get less risk-averse as number of iterations dropped, approaching expectation maximization. This would reflect the idea that the Kelly criterion arises because of long-term performance over many iterations, and normal expectation maximization is the right thing to do in single-shot scenarios.
But I sort of suspect this "origin of Kelly" story is wrong. So I'm also interested in number-iteration formulas which reach different conclusions.
The obvious route is to modulate Kelly by the probability that the result will be close to the median case. With arbitrarily many iterations, we are virtually certain that the fraction of bets which pay out approaches their probabilities of paying out, which is the classic argument in favor of Kelly. But with less iterations, we are less sure. So, how might one use that to modulate betting behavior?
I suggest explicitly stepping outside of an expected-utility framework here. The classic justification for Kelly is very divorced from expected utility, so I doubt you're going to find a really appealing generalization via an expected-utility route.