So crops carry over from one month to the next, those ones that will regrow, or do we need to replant each and every month?
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Two crops regrow, cucumber and corn. At least cucumber indicate they regrow on their own in its description for them anyway. Just like corn does.
Two do not, turnip and potato, IIRC.
Seeing as how we are in this for the long game anyway seeing as how it takes the time it does to navigate through all of the NPC's events, I figure why not just plant all of the crops that regrow on their own to begin with seeing as how there are no recipes that require turnips or potatoes or no NPC's request them either.
Even if they take longer to initially mature, if they continue to fruit forever, you're better off in the long run going this route to begin with, I figure.