There are versions of the VNM theorem that allow infinitely many possible outcomes, but they either

1) require additional continuity assumptions so strong that they force your utility function to be bounded

or

2) they apply only to some subset of the possible lotteries (i.e. there will be some lotteries for which your agent is not obliged to define a utility).

I might look it up on the original paper when I have time.

The original statement and proof given by VNM are messy and complicated. They have since been neatened up a lot. If you have access to it, try "Follmer H., and Schied A., *Stochastic Finance: An Introduction in Discrete Time*, de Gruyter, Berlin, 2004"

EDIT: It's online.

Also, it's well possible that your utility function doesn't evaluate to +10000 for any value of its argument, i.e. it's bounded above.

Since utility functions are only unique up to affine transformation, I don't know what to make of this comment. Do you have some sort of canonical representation in mind or something?