Totally depends on what you mean when you use the word "master".
If you mean achieving perfect skill and understanding of a style, then the answer is zero. No one ever reaches perfection.
If you mean achieving sufficient skill and understanding of the official curriculum of an art (assuming it has one) to be considered a competent instructor, then it depends on:
- what level of expertise you consider the minimum for a "competent" instructor
- how much time and energy the individual has available to devote to their study
- how talented they are at learning a martial art
- how efficiently they use their available training time and how effective their teachers are at transmitting the knowledge and skills they need to learn
Let's imagine that you consider a practitioner of average skill should be able to reach the level of competent instructor after 5,000 hours of training in a given art. If that person started at age 12 and trained 20 hours per week until they reached age 72, then they would have 62,400 hours of training. That comes out to 12.5 arts trained to instructor level. However, you have to account for time spent maintaining skills and knowledge, otherwise the practitioner would have forgotten the first arts learned by the time he/she got to the later ones. Let's set aside 12,400 hours for just maintaining skills already learned. That drops the total number of arts "mastered" by age 72 down to 10.
There's been a lot of buzz about the idea (promulgated by Malcolm Gladwell) that 10,000 hours of focused practice is what you need to reach world-class expertise in a given field. If you accept that as your standard, then our hypothetical practitioner could have reached mastery in 5 arts by age 72.
One thing not always mentioned about that 10,000 hours theory is that it's based on the idea of focused, mindful practice with a clear aim for improvement in each practice session, not just mindless repetition. If the practitioner has a lower quality of practice than that, then the number of arts drops further still - potentially down to zero. A lifetime of practice badly done may not ever produce mastery.
On the other hand, consider a practitioner who is unusually efficient in organizing their practice sessions, has remarkable natural talent, and takes proper advantage of the lessons learned in one art when approaching the next. Proper advantage does not mean trying to force one art into the mold of another. It does mean understanding which principles overlap and making use of generalizable attributes like kinesthetic awareness, balance, sensitivity and mental fortitude. In this case, perhaps the lifetime number would jump back up to 10. If the arts studied are closely related, it may go even higher. Arts like Judo, Sambo, and BJJ are closely related. Someone with 10,000 hours of dedicated Judo study will not need nearly that much time to reach a comparable level when starting over in BJJ.
That last issue deserves some attention. Some people, myself among them, would argue that Judo, BJJ, Sambo, Catch Wrestling, and similar systems are just different aspects of the same art optimized for specific contexts or competition rule sets. From this perspective, someone who is really good at BJJ, Sambo, Catch, and Judo isn't necessarily a master of 4 different arts - they're a master of grappling who happens to know how to apply that skill in a variety of settings. I'm sure there are other groupings of arts out there which are similar - common principles but different names based on political splits or specialized optimization for a particular context. Someone who works to "master" arts within one of those groupings could achieve respectable proficiency much more quickly than someone who practiced a more diverse selection of arts.
