Say you're on your custom demiplane with a custom magic item that let's you push a button to expand your demiplane.
Say you go into Time Stop and mash that magic button like there's no tomorrow.
Do the castings pile up and all go off at once or no?
Why and why not? And are there other spells or precedent for such?
Assuming it works are there other spells with similar wording that might be similarly useful?
Pushing a button (that has a major magical effect) is going to be a standard action in PF. Seriously, it's a 9th level spell with a casting time of 6 hours, it's just not going to work as a free action a hundred times a round. So yes, you can expand your demiplane quickly with your awesomely expensive custom magic item and time stop.
Though I can't see how that combo's useful, just create your demiplane in advance and time stop is irrelevant here. Even the custom magic item isn't really required.
I think they might be trying to find a way to effectively have a permanent demiplane without paying for it.
since create demiplane greater has a duration of days per level. It seems like it'd be easy enough to just recast it once every week or so, if you don't feel like paying for it. If you're worried that you can't recast it once every 2 weeks you could get a metamagic rod of extend greater and now you only need to recast it once a month.
Which you're already doing via timestop and said expensive custom magic item.
As for the OP question. Each casting would add to the demi-plane and each expansion would have a duration of 1 day per level that would wink out once the duration ends. As for why, it's right in the description of the spell.
If the duration on one area of the demiplane ends and other parts remain, creatures in the expiring area are shunted to remaining areas. If a collapsing portion of the demiplane would leave one section cut off from other sections of the demiplane (for example, if there were three areas connected in a straight line and the center part expired), the stranded sections count as separate demiplanes under your control. You may reconnect these stranded sections by casting the spell again to create a linked area between the two.
Also, unless specified otherwise activating such a magic item would take 6 hours. This means that you would still be in the process of activating it once time stop wore off.
Activating a magic item is a standard action unless the item description indicates otherwise. However, the casting time of a spell is the time required to activate the same power in an item, regardless of the type of magic item, unless the item description specifically states otherwise.
Even if this item only took a standard action to activate, Time Stop isn't doing anything for you that you couldn't do with out it. You could simply mash the button without time stop and rapidly expand the demi plane.
I agree with AVR I don't see how this is useful so it's hard for me to give examples of "other spells that would be similarly useful with timestop"