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TimFreeman comments on A summary of Savage's foundations for probability and utility. - Less Wrong
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Okay, what field do you think works for utility that's better than real numbers?
The obvious candidates are surreal numbers or non-standard reals. Wikipedia says that the former doesn't have omega plus 1, where omega is the number of ordinary integers, but IIRC the latter does, so I'd try the latter first. I do not feel confident that it solves the problem, though.
The surreals do have ω+1 - see the "..And Beyond" section of the wiki page. If this is contradicted anywhere else on the page, tell me where and I'll correct it.
The surreals are probably the best to use for this, though they'll need to emerge naturally from some axioms, not just be proclaimed correct. From WP: "In a rigorous set theoretic sense, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals.", so even if the surreals are not necessary, they will probably be sufficient.
Conway used surreal numbers for go utilities. I discussed the virtues of surreal utilities here.
Those aren't really utilities because they aren't made for taking expectations, though any totally ordered set can be embedded in the surreals, so they are perfect for choosing from possibly-infinite sets of certain outcomes.
Checking with the definition of utility expectations do not seem critical.
Conway's move values may usefully be seen as utilities associated with possible moves.
Okay, that is not the kind of utility discussed in the post, but it is still a utility.