1. This should properly be broken down into two questions: the most-flown model, or low-power, rocket, and the most-flown high-power rocket. Two entirely different challenges, both logistically and financially.
2. The meaningful answer involves a precise definition of what you mean by "a rocket". Do you mean precisely the same set of parts, shock cords, parachutes, nose cones, body tubes, and paint job, or do you mean an assembly of parts that all look the same and have the same name, but have been through the sort of real-life attrition that affects all of us. This reminds me of that old chestnut about the old carpenter that claimed he was still using his great-grandfather's hammer. Since his ggf had owned it, it had seven new handles and two new heads, but it was still "his great-grandfather's hammer". I still have the "same rocket" that I certified L1 with back in 1998, and it has flown 146 times, but there is not a single molecule on this rocket that rode along on that first flight. I still keep the same name because it is painted exactly the same, and it still has a 38mm motor mount tube. It's called the "Astro*Mollusc". Therein lies a tale.
3. As far as high-power records for the most flights go, a serious contender must be a tube-finned rocket called "Tuber" that a member of the MDRA group has been flying since the biblical floods of Noah receded from the face of the earth. I have been racking my brain to remember his last name, Dave somebody, but this rocket has been flying forever, and, as far as I know, is still in the "church militant'. It has probably flown 300 times.
4. Most people just don't keep the kind of records that would allow a question like this to be answered with any sort of reliability, so this will have to remain in the realm of 'urban legend".
Alan