Presumably, if Utilions are useful at all, then you use them. Usually, this means that some are lost each day in the process of using them.
Further, unless the Utilions represent some resource that is non-entropic, then I will lose some number of Utilions each day even if they aren't lost by me using them. This works out to the same answer in the long run.
Let's assume we have an agent Boxxy, an immortal AI whose utility function is that opening the box tomorrow is twice as good as opening it today. Once he opens the box, his utility function assigns that much value to the universe. Let's assume this is all he values. (This gets us around a number of problems for the scenario.)
Even in this scenario, unless Boxxy is immune to entropy, some amount of information (and thus, some perception of utility) will be lost over time. Over a long enough time, Boxxy will eventually lose the memory of opening the Box. Even if Boxxy is capable of self-repair in the face of entropy, unless Boxxy is capable of actually not undergoing entropy, some of the Box-information will be lost. (Maybe Boxxy hopes that it can replace it with an identical memory for its utility function, although I would suspect at that point Boxxy might just to decide to remember having opened the Box at a nearer future date) Eventually, Boxxy's memory and thus, Boxxy's Utilions, will either be completely artifiical with at best something like a causal relationship to previous memory states of opening the box, or Boxxy will lose all of its Utilions.
Of course, Boxxy might never open the box. (I am not a superintelligence obsessed with box opening. I am a human intelligence obsessed with things that Boxxy would find irrelevant. So I can only guess as to what a box-based AGI would do.) In this case, the Utilions won't degrade, but Boxxy can still expect a value of 0 in this case.
Frankly, the problem is hard to think about at that level, because real immortality (as the problem requires) would require someway to ensure that entropy doesn't occur but somehow some sort of process occurs, which seems a contradiction in terms. I guess this could be occuring in a universe without entropy, (but which somehow has other processes) although both my intuitions and my knowledge are so firmly rooted in a universe that has entropy that I don't have a good grounding on how to evaluate problems in such a universe.
Presumably, if Utilions are useful at all, then you use them. Usually, this means that some are lost each day in the process of using them.
No. Those are resources.
Let's say you have a box that has a token in it that can be redeemed for 1 utilon. Every day, its contents double. There is no limit on how many utilons you can buy with these tokens. You are immortal. It is sealed, and if you open it, it becomes an ordinary box. You get the tokens it has created, but the box does not double its contents anymore. There are no other ways to get utilons.
How long do you wait before opening it? If you never open it, you get nothing (you lose! Good day, sir or madam!) and whenever you take it, taking it one day later would have been twice as good.
I hope this doesn't sound like a reductio ad absurdum against unbounded utility functions or not discounting the future, because if it does you are in danger of amputating the wrong limb to save yourself from paradox-gangrene.
What if instead of growing exponentially without bound, it decays exponentially to the bound of your utility function? If your utility function is bounded at 10, what if the first day it is 5, the second 7.5, the third 8.75, etc. Assume all the little details, like remembering about the box, trading in the tokens, etc, are free.
If you discount the future using any function that doesn't ever hit 0, then the growth rate of the tokens can be chosen to more than make up for your discounting.
If it does hit 0 at time T, what if instead of doubling, it just increases by however many utilons will be adjusted to 1 by your discounting at that point every time of growth, but the intervals of growth shrink to nothing? You get an adjusted 1 utilon at time T - 1s, and another adjusted 1 utilon at T - 0.5s, and another at T - 0.25s, etc? Suppose you can think as fast as you want, and open the box at arbitrary speed. Also, that whatever solution your present self precommits to will be followed by the future self. (Their decision won't be changed by any change in what times they care about)
EDIT: People in the comments have suggested using a utility function that is both bounded and discounting. If your utility function isn't so strongly discounting that it drops to 0 right after the present, then you can find some time interval very close to the present where the discounting is all nonzero. And if it's nonzero, you can have a box that disappears, taking all possible utility with it at the end of that interval, and that, leading up to that interval, grows the utility in intervals that shrink to nothing as you approach the end of the interval, and increasing the utility-worth of tokens in the box such that it compensates for whatever your discounting function is exactly enough to asymptotically approach your bound.
Here is my solution. You can't assume that your future self will make the optimal decision, or even a good decision. You have to treat your future self as a physical object that your choices affect, and take the probability distribution of what decisions your future self will make, and how much utility they will net you into account.
Think if yourself as a Turing machine. If you do not halt and open the box, you lose and get nothing. No matter how complicated your brain, you have a finite number of states. You want to be a busy beaver and take the most possible time to halt, but still halt.
If, at the end, you say to yourself "I just counted to the highest number I could, counting once per day, and then made a small mark on my skin, and repeated, and when my skin was full of marks, that I was constantly refreshing to make sure they didn't go away...
...but I could let it double one more time, for more utility!"
If you return to a state you have already been at, you know you are going to be waiting forever and lose and get nothing. So it is in your best interest to open the box.
So there is not a universal optimal solution to this problem, but there is an optimal solution for a finite mind.
I remember reading a while ago about a paradox where you start with $1, and can trade that for a 50% chance of $2.01, which you can trade for a 25% chance of $4.03, which you can trade for a 12.5% chance of $8.07, etc (can't remember where I read it).
This is the same paradox with one of the traps for wannabe Captain Kirks (using dollars instead of utilons) removed and one of the unnecessary variables (uncertainty) cut out.
My solution also works on that. Every trade is analogous to a day waited to open the box.