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A question about Ripple Effect (AKA Ripple Play)

I don't know if Erich Friedman's Math Magic allows user-submitted problems (otherwise, there might be way more problems than just one per month), but this problem is really nagging at me, having written some Ripple Effect puzzles recently for Grandmaster Puzzles. (Will they be used? Time will tell.)

What's the smallest rectangular Ripple Effect puzzle with a unique solution and no givens such that there's at least one region of size n? Does such a puzzle exist for every n? Here are examples for n=1,2,3,4,5:

Special thanks to Luke Pebody for supplying the solution for n=4, and Bryce Herdt for a solution for n=5 (which was obsoleted by Luke, but nonetheless helpful). What about for n>6?

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