Does the game have any sort of plot, that can eventually be finished/completed, or is it just the sandbox to go at forever?

If you go to northwest corner of the farm, there is some slight hint of there being a plot (or more exactly non-sandbox element, I think the game will just continue after that hinted event). Otherwise, it's pure sandbox. Start with almost nothing and constantly improve your tools and possibilities, until you collect everything, befrient everyone and generally stop having fun. is the same as in Terraria or Recettear - immense timewaster, and I'm enjoying every minute of it (reached my first winter yesterday, finally I can upgrade my watering can in peace, though I suspect I will be using it much less in the second year).

The game gives you a quest, besides the quests you receive in mails and in the town (like the papers in the walls you can read for optional quests, and community center bundles to help Junimos).
I guess you could call it the main quest, considering who gives you that mission.
The good news is that even if you fail, the game allows you to repeat. You don't need to restart the game from zero to obtain success in the quest.
It wasn't always like this, but after fans complained, ConcernedApe added the option to repeat :P

Do you repeat from the first year in an alternate timeline while keeping your stuff, or do you simply continue on with no deadline? I know that the ghost visits the first day after Year 2 to evaluate your status, but that's all I know. I think you can watch TV in the morning to get the following day's weather. If it's Day 1, and it's going to rain on Day 2, then you water your crops Day 1 and give Clint your watering can before 4pm. On Day 2, it rains. On Day 3, you get your upgraded can, and then you water your plants before the end of the day. I think that'll work, but I haven't done it yet.

It's a good thing weather forcasting is a 100% accurate science. (I still think the upgrading system is a bit obnoxious and unnecessary...)

I tried to keep this spoiler-free... I think you just go on, you even have the option to redo the evaluation if you don't get max points. But I have yet to experience it myself. I knew about the TV and I got my watering can up to steel this way... but it either stubbornly refused to rain in the fall, or there was a festival in the way - and Clint doesn't work on festival days. Which means one day delay for growing crops, and all plants scheduled to harvest at 28th now get one less harvest. By the way I have suspicion that the forecast is not 100% accurate. Even disregarding possibility of rain totem, I am almost sure I saw nondescript "windy and cloudy" forecast and the next day a storm came instead.

Anyway all that is thing of the past now... Just as I suspected, I got enough material over that winter to use sprinklers exclusively for the next year. The gold watering can now rusts (or whatever it is that gold does) in my "planting supplies" chest and gets some use only on the first few days of a season, when tilling new plots. Right now I'm in the middle of second winter, and looking forward to get my special reward.

[Spoilers]
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Grandpa quest was changed in patch 1.05d. Here is what ConcernedApe said: You can re-summon him any time, as long as you have a diamond, so you don't need to restart game and lose your items, relationship with villagers, etc.
So don't worry if you are near year 3 and thinks you haven't done enough to pass the test ˆˆ
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[More spoilers]
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When you complete the quest, he will give a statue (Statue of Perfection) which generates Iridium ore every day.
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[Spoilers]

Nice info, thanks. Of course, if you've already collected All The Things and Won The Game, you probably don't even need the ores. Sort of like how by the time you can max your watering can, you should have an array of super sprinklers anyway...

Some of these bundles are harsh, though. I might not complete any of the fishing bundles until Year 2, if ever. You need a second computer just to have the wiki open, and even then, it could take several days to get the right fish. (Or maybe better gear and skills changes that.)