200. Obviously, you would have to change the last digit, but 201, 203, 207, and 209 are all composite. For any smaller number, you can change the last digit, and get 2,11,23,31,41,53,61,71,83,97,101,113,127,131,149,151,163,173,181, or 191.

200+2310n gives an infinite family, because changing the last digit to 1 or 7 gives a number divisible by 3; to 3, a number divisible by 7; to 9, a number divisible by 11.