I guess my question would be does your consecutive rack run have to have been accomplished in a tournament match or gambling session, or do practice sessions / casual games with no wager count as well? I was kind of feeling it has to be the former, however for years when people have asked me what my high run is in 14.1, I consider my high run accomplished during solo practice as my high run. So based on that, it would seem inconsistent for me to consider that a personal best consecutive rack run in 9-ball should only count when executed in a tournament match or gambling session. So I guess I'd say you should consider both as separate accomplishments - your PR consecutive rack run in a practice / casual session as well as your PR consecutive rack run in a tournament / gambling session. There certainly is a difference.
The only hole-in-one I've ever had in golf was accomplished during a round playing by myself, non-witnessed, so when anyone has asked me if I've ever had a hole-in-one, I tell them no, unless I decide to elaborate and tell them about that one I had when playing by myself, which I don't really count.
To answer the question, for 9-ball in a match or tournament, probably 4 racks, but honestly I'm not sure. At least for myself which I'm guessing is common for others, when you get in one of those zones of playing really well and stringing racks, it just happens and you're not concerned with counting how many consecutive racks you may have just run, so unless an observer happens to notice it and let you know, you possibly may not even realize it.