When it comes to gaining more than one Ph.D., one is going to encounter some issues. Most institutions will not allow a second Ph.D. unless it can be proven that the fields are far enough apart, and will benefit the one seeking the secondary Ph.D. When searching for who holds the most doctoral degrees, most hold only one, unless you are counting honorary doctoral degrees, which technically do not count as “having earned” a Ph.D. With the honorary doctoral degrees, the winning number would go to the King of Thailand, who holds ten honorary doctoral degrees.
There are reports of some German and French professors holding multiple Ph.Ds, but there have not been any corroborating evidence shown, and when research is done, the search comes up empty. Many self-promoters such as Benjamin Bolger claim to hold multiple doctoral degrees, but again, there is no corroborating evidence to show these numbers are accurate.