"These are simply problems that have no solutions, like the problem of finding the largest integer has no solution. You can get arbitrarily close, but that's it." - Actually, you can't get arbitrarily close. No matter how high you go, you are still infinitely far away.

"How can utility be maximised when there is no maximum utility? The answer of course is that it can't."

I strongly agree with this. I wrote a post today where I came to the same conclusion, but arguably took it a step further by claiming that the immediate logical consequence is that perfect rationality does not exist, only an infinite series of better rationalities.