A few comments on the remainder of this subthread.
Different people are talking about different things, without making the distinctions perfectly clear. Specifically, when Lumifer says "it maximizes your expected bankroll, and maximizing your expected log bankroll is the same" I believe he means "it maximizes your expected bankroll in the long run, and maximizing your expected log bankroll in the long run is the same because in the long run you almost always get approximately your expectation". Whereas DanielLC and I have been observing that on a single iteration the Kelly rule maximizes your expected log bankroll and does not maximize your expected bankroll. (But Lumifer hasn't been saying "in the long run" every time, and we haven't been saying "on a single bet" every time, hence a lot of what I think is pseudo-disagreement.)
It is simply not true in general that maximizing E(X) is the same as maximizing E(f(X)) when f is a monotone increasing function. It is true when X is not actually random, which is kinda what happens in the long run here.
It is simply not true that the Kelly rule "assumes logarithmic utility of money". If your utility happens to be proportional to log(money) then the Kelly rule maximizes your expected utility even for a single bet, but provided your dependence of utility on money is monotone increasing, in the long run your utility will almost always be maximized by following the Kelly rule.
It isn't clear to me that the only reason why maximizing E(X) and maximizing E(log(X)) are different is that "zero is special", even when we are considering what happens in the long run. Specifically, suppose your individual bets have some nasty distribution whose tails are too fat for the variance to be defined; then it needn't be true that your performance almost always looks like its expectation. Isn't there a counterexample lurking somewhere around here?
I think it's a distraction to talk about what the Kelly rule is "trying" to do. You can say that it almost always maximizes your bankroll in the long run. You can say that it maximizes your expected utility in the short run, if its dependence on wealth is logarithmic. You can say that it gives you the warm fuzzy feeling of doing something known to be optimal. The Kelly rule doesn't care; it is what it is, whatever your goal is. And what it is is an injunction to choose each individual bet so as to maximize E(log(bankroll)).
The principle of maximizing E(log(bankroll)) can be applied in situations where the choice you have is something other than simply "how much shall I bet on this otherwise-fixed gamble?". I don't think it's a serious abuse of terminology to call this "the Kelly rule" since it's simply a generalization of Kelly to this broader class of situations.
[EDITED to fix formatting a bit.]
It isn't clear to me that the only reason why maximizing E(X) and maximizing E(log(X)) are different is that "zero is special", even when we are considering what happens in the long run. Specifically, suppose your individual bets have some nasty distribution whose tails are too fat for the variance to be defined; then it needn't be true that your performance almost always looks like its expectation.
In particular its possible for log(X) to have well-defined variance but not X, and for E(log(X)) but not E(X) to be defined.
A lottery ticket sometimes has positive expected value, (a $1 ticket might be expected to pay out $1.30). How many tickets should you buy?
Probably none. Informally, all but the richest players can expect to go broke before they win, despite the positive expected value of a ticket.
In more precise terms: In order to maximize the long-term growth rate of your money (or log money), you'll want to put a very small fraction of your bankroll into lotteries tickets, which will imply an "amount to invest" that is less than the cost of a single ticket, (excluding billionaires). If you put too great a proportion of your resources into a risky but positive expected value asset, the long-term growth rate of your resources can become negative. For an intuitive example, imagine Bill Gates dumping 99% percent of his wealth into a series of positive expected-value bets with single-lottery-ticket-like odds.
This article has some graphs and details on the lottery. This pdf on the Kelly criterion has some examples and general dicussion of this type of problem.
Can we think about Pascal mugging the same way?
The applicability might depend on whether we're trading resource-generating-resources for non-resource-generating assets. So if we're offered something like cash, the lottery ticket model (with payout inversely varying with estimated odds) is a decent fit. But what if we're offered utility in some direct and non-interest-bearing form?
Another limit: For a sufficiency unlikely but positive-expected-value gamble, you can expect the heat death of the universe before actually realizing any of the expected value.