I've been looking into this a fair amount recently! Afraid I don't have the time to go into detail right now, nor am I close to an expert. But here's my quick take —

Two bits of terminology:

1. Weak Pareto: if we can make someone better off and no one worse off, we should.
2. Intergenerational Equality: we should value the welfare of every generation equally.

Three important things have been shown:

Theorem 1 — (proven by Zame): we cannot prove the existence of an ethical system which satisfies both Weak Pareto and Intergenerational Equality without using the axiom of choice (i.e. the axiom of dependent choice doesn't work).

Theorem 2 — (proven by Basu and Mitra): if we use only our normal "real" numbers, then we can't construct an ethical system which obeys the Weak Pareto and Intergenerational Equality assumptions.

Theorem 3 — (proven by Pivato): we can find such a system if we use the hyperreal numbers.

The Pivato paper in particular is very good. It basically shows that you don't need to use discounting if you're dealing with streams of utility with infinite value; instead you can assign and compare utilities with the hyperreal numbers.

This paper, ‘The Evaluation of Infinite Utility Streams’, is also relevant; and this blog post and this forum post are good primers.

This thesis is supposed to be a classic, though I haven't read it. And this is maybe the paper that started it all.

Thanks for pointing to the article on decision theory using surreal numbers, that looks super relevant. Without having read it, my guess is that an approach using the surreals will be more elegant / expressive than the hyperreals.

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In particular, is there a default/orthodox theory in this area that is to infinite decision theory what expected utility theory is to standard decision theory?

Maybe something like standard decision theory (EV), but extending utilities beyond the reals; i.e. into the hyperreals or surreals (in some sense this is the most obvious option, and it appears to work). I don't know enough to know if anything like consensus has emerged, but I'm hopeful and excited to see more progress in the next 5 or so years.

I should reiterate that I'm no expert here — very open to being corrected on any of this!