There are a finite number, in fact, getting them all is an achievement. They cannot be repeated
They can be found:
- at the Stardew Valley Fair for 2000 tokens
- on floor 100 of The Mines.
- by getting your spouse to "13"/12 hearts
- bought from Krobus for 20000g in the Sewers.
- give the statue in the secret woods a Sweet Gem Berry
- getting the Master Angler Achievement
There are two Stardrops that are contested:
Some users swear up and down that there is an additional Stardrops for giving your spouse a birthday present, while others insist that's just the 13/12 one. (I personally got the affection one, then gave Haley a coconut oh her birthday two years in a row and did not receive an additional Stardrop)
Supposedly completing the museum collection awards a Stardrop, but it's possible to get 12/13 honors without completing the museum, with the missing one being "get all the achievements." "Get all Stardrops" is also an honor, so if there IS one from the museum, it doesn't count towards the score at the end.