2519, and the Shift That Turns Nine Congruences Into One
Every one of the nine conditions leaves a remainder one short of its divisor, so x plus one is divisible by all of 2 through 10 and the answer is 2519. Minimality comes free, and the whole solution set is 2520k minus 1. The article carries the coprimality caveat, the near miss 209 that satisfies six of the nine, and a variant where no shift exists.