What is the smallest number that cannot be made prime by changing a single digit?

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.