The Kelly criterion assumes logarithmic utility of money. The difference between $100 and $10 is the same as the difference between $10 and $1. And going broke has negative infinite expected value. If you don't do that, then your best bet is to take any bet with expected value above one. Sure you'll almost certainly go broke, but the utility from winning every bet would be enough to make up for it.
Even if we assign logarithmic utility to whatever we're being mugged with, it just makes getting mugged harder. 3^^^3 is such overkill that, unless we're running out of money and we assign negative infinite utility to this, we're better off paying the mugger.
A slight error: the Kelly criterion assigns no utility. It simply maximizes your expected bankroll at any point in the non-immediate future. What you do with that bankroll and how much you care about it is unmentioned by the Kelly criterion.
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.