Does it occur every time?

If $\overline{ABC}=100$ , then the largest prime divisor of $\overline{ABCABCABC}$ is $333667$ . So the answer is at most $333667$ . If we try with other numbers, we will experience that $333667$ is probaly the answer. So let's prove it!

Let's see $\overline{ABCABCABC}$ in an other way $\overline{ABCABCABC}=\overline{ABC000000}+\overline{ABC000}+\overline{ABC}=\overline{ABC}*10^6+\overline{ABC}*10^3+\overline{ABC}={\color{#D61F06}{1001001}}*\overline{ABC}$ Since $1001001=3*333667$ , where both of $3$ and $333667$ is prime, $\overline{ABCABCABC}$ is definitely divisible by $333667$ .

Therefore the answer is $\boxed{333667}$ .