Forget the "greatest good for the greatest number" part, it's irrelevant to the question. Would you prefer to have the same fun sooner, rather than later, and how much time would you trade in delay vs duration?

The answer is, as always, "it depends." Seriously , though- I time discount to an extent, and I don't want to stop totally. I prefer more happiness to less, and I don't want to stop. (I don't care about ending date, and I'm not sure why I would want to). If a trade off exists between starting date, quality, and duration of a good situation, I'll prefer one situation over the other based on my utility function. A better course of action would be to try and get more information about my utility function, rather than debating which value is more sacred than the rest.

My guess is the second greatest good would be the best option. Taking into account that these are mortal beings and you don't know how long will these people live/be healthy, the best choice would be one that is not the least rewarding (first greatest good) neither the slowest to come.

Think of it as spending your life savings: you don't want to spend it all on your youth and be poor the rest of your life and you don't want to spend it all in your nineties either, when you'll be too old to enjoy it. The answer is somewhere in the middle.

This post is far too unclear and ill-defined to even comment meaningfully on. It reminds me of TimeCube. Downvoting.

What within utilitarianism allows for selecting between these three greatest good for the greatest number?

I don't think there's anything baked into Millian utilitarianism that automatically says what to do here, but then (1) there are other kinds of utilitarianism and (2) bolting the notion of time discounting onto Millian utilitarianism ought to solve the question. Compute the time integral of time-discounted goodness for each of the three greatest goods over all time, and pick the greatest good that gives the most positive answer. [Dusts hands.]

I think you may have some typos such as 'for the greatest number for the greatest number' in paragraph 1 and 'will start end' in paragraph 2. But that aside, if I throw in some concrete numbers:

GG1: 1 year of fun, starting today.

GG2: 1 year of fun, starting 3 years from now and ending 4 years from now.

GG3: 2 years of fun, starting 1 year from now and ending 3 years from now.

If you were to heavily time discount, you would probably pick GG1.

If you were to simply want most person years of fun, you would probably pick GG3.

If you were under the impression that this was heavily focused on a survival analysis (Dr. Dystopia, unless stopped, will cause absolutely no effects until the fun starts, and then at the end of the fun period will exterminate everyone forever.) then you might want to pick GG2, since that gives the most time to come up with a plan to stop Dr. Dystopia.

If three people all have comparable utility beliefs except that one is heavily time discounting, one is heavily valuing person years and one is heavily thinking of survival analysis and they need to vote on those scenarios, presumably they have different priors and can begin discussing the various types of evidence they have for those beliefs and can attempt to come to an accurate conclusion.

Does that help? I'm a bit concerned I'm not addressing the core question, but I'm not sure what else to say yet.