I got it, sam.
Doesn’t help much, does it? 
Well, I don’t want to brag, but I’ve had 33 years of Fish, and so far it’s been pretty good.
Wait, which Fish are we talking about?
I thought the OP was about hand-me-down names. Y’know, the Sr. Jr. and whatnot.
What family has the longest run of Charles Snottington XXI ?
Personally, a friend of a friend, if their next child is a boy, will be Paul (Lastname) the V. That is the highest personal roman numeral person I will know of.
Well, if we can count papal names, the highest is somewhere around the upper teens, IIRC. 
Just in case anyone is unsure, Adamses and Fishes are the correct plurals.
Well, a classmate of mine had a son a few years ago, and gave him a “the Fifth” name. He calls the kid by his middle name. The rest of us call him “Mk. 5” or “Version 5.0”.
And popes beat the high teens… It wasn’t all that long ago (four popes ago, I believe) that the Pope was John XXIII.
How dare you! 