After considering this problem, what I found was surprisingly fast, the specifics of the boxes physical abilities and implementation becomes relevant. I mean, let's say Clippy is given this box, and has already decided to wait a mere 1 year from day 1, which is 365.25 days of doubling, and 1 paperclip is 1 utilon. At some point, during this time, before the end of it, There are more paperclips then there used to be every atom in the visible universe. Since he's predicted to gain 2^365.25 paperclips, (which is apparently close to 8.9*10^109) and the observable universe is only estimated to contain 10^80 atoms. So to make up for that, let's say the box converts every visible subatomic particle into paperclips instead.
That's just 1 year, and the box has already announced it will convert approximately every visible subatomic particle into pure paperclip bliss!
And then another single doubling... (1 year and 1 day) Does what? Even if Clippy has his utility function unbounded, it should presumably still link back to some kind of physical state, and at this point the box starts having to implement increasingly physically impossible ideas to have to double paperclip utility, like:
Breaking the speed of light.
Expanding the paperclip conversion into the past.
Expanding the paperclip conversion into additional branches of many worlds.
Magically protecting the paperclips from the ravages of time, physics, or condensing into blackholes, despite the fact it is supposed to lose all power after being opened.
And that's just 1 year! We aren't even close to a timeless eternity of waiting yet, and the box already has to smash the currently known laws of physics (more so than it did by converting every visible subatomic particle into paperclips) to do more doublings, and will then lose power afterwards.
Do the laws of physics resume being normal after the box loses power? If so, massive chunks of utility will fade away almost instantly (which would seem to indicate the Box was not very effective), but if not I'm not sure how the loop below would get resolved:
The Box is essentially going to rewrite the rules of the universe permanently,
Which would affect your utility calculations, which are based on physics,
Which would affect how the Box rewrote the rules of the universe,
Which would affect your utility calculations, which are based on physics,
Except instead of stopping and letting you and the box resolve this loop, it must keeps doubling, so it keeps changing physics more.
By year 2, it seems like you might be left with either:
A solution, in which case whatever the box will rewrite the laws of physics to, you understand and agree with it and can work on the problem based on whatever that solution is. (But I have no idea how you could figure out what this solution would be in advance, since it depends on the specific box?)
Or, an incredibly intractable metaphysics problem which is growing more complicated faster than you can ever calculate, in which case you don't even understand what the box is doing anymore.
The reason I said that this was incredibly fast is that my original guess was that it would take at least 100 years of daily doubling for the proposed world to become that complicated, but when I tried doing a bit of math it didn't take anywhere near that long.
Edit: Fixed a few typos and cleared up grammar.
Utility doesn't have to be proportional to the amount of some particular kind of physical stuff in the universe. If the universe contained 1 paperclip, that could be worth 2 utilons, if it contained 2 paperclips then it could be worth 4 utilons, if it contained 20 paperclips then it could be worth 2^20 utilons. The box would then double your utility each day just by adding one physical paperclip.
I still think these kinds of considerations are worth thinking about though. Your utility function might grow faster than a busy beaver function, but then the doubling box is going to have trouble waiting the right length of time to deliver the
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.