The longest chain of the same personal name used from father to son from generation to generation that I know of seems to be at least 22 generations according to my sources.
One example of a long chain of using the same personal name is: Louis XVII of France, the Dauphin in the Temple, was the son of Louis XVI, the son of Dauphin Louis, the son of Louis XV, the son of Louis the Small Dauphin, the son of Louis the Grand Dauphin, the son of Louis XIV, the son of Louis XIII. That's eight generations of Louis in a row.
But that is nothing compared to the longest chain of same name fathers and sons known to me.
In Germany one of the important noble families is that of Reuss. And like most noble families they started out in history rather small and got more important over time. And to honor a benefactor who made an ancestor more important, every male born into the Reuss family has been named Heinrich (Henry) since the reign of Emperor Henry VI.
To be precise, the person who historians list as Emperor Henry VI was actually Emperor Heinricus V, King Heinrich VI of Germany, King Enrico V of Italy, King Henri IV of Burgundy, & King Enrico I of Sicily.
And his reign was from 1190 to 1197.
So all male babies born into the Reuss dynasty have been named Heinrich for over 820 years. So every Heinrich in the Reuss family knows that his brothers are named Heinrich, his uncles and cousins are named Heinrich, his father is named Heinrich, and his sons and grandsons will be named Heinrich.
And I suppose even a very young boy of the Reuss family can recite his agnatic (male only) ancestry for many generations or centuries into the past. "Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, son of Heinrich, ....
So a new born Heinrich of Reuss, born about 820 years since his family started using Heinrich as the only male name, could have about twenty straight generations of Heinrich in his ancestry. And that is if the average generation length was forty years, which seems way too long over twenty generations. The actual figure might be about thirty generations of Heinrich.
So you need to find a pedigree which shows all the generations named Heinrich in the Ruess family. If you can find one that will be the record to beat for the family with the longest succession of the same name.
Added November 02, 2019.
In this site http://www.angelfire.com/realm/gotha/gotha/reuss.html1 I find a prince Henrich VII (b. 2017) son of Prince Heinrich XIX (b. 1974), son of Heinrich VII (1927-2002), son of Heinrich XXXIX (1891-1946), son of Heinrich XXIV (1855-1910), son of Heinrich IV (1821-1894), son of Heinrich LXIII (1786-1841), son of Heinrich XLIV of the Junior Line (1753-1832).
So that is eight generations of Heinrich born during a period of about 264 years. With seven gaps between the 8 generations that makes an average of 37.714 years per generation. That indicates there might have been 21 or 22 generations of Heinrich in the Ruess dynasty.
Heinrich XLIV of the Junior Line (1753-1832) was the son of Heinrich IX, Count Reuss of Kostritz (1711-1780), the son of Heinrich XXIV, Count of Reuss-Schleiz-Kostritz (1681-1748), son of Heinrich I, Count of Reuss-Schleiz (1639-1692). https://en.wikipedia.org/wiki/Heinrich_LXIII,_Prince_Reuss_of_K%C3%B6stritz#Ancestry2
So that makes 11 generations, or 10 gaps between generations, born in 378 years from 1639-2017, with an average of 37.8 years per generation. Thus at that rate there could have been 21 or 22 generations with the same name born in 820 years.
Heinrich I, Count of Reuss-Schleiz (1639-1692), was a son of Heinrich III (1603-1640), son of Heinrich (1572-1635), son of Heinrich XVII (1530-1572), son of Heinrich XIV, the last Vogt of Plauen (c. 1464-1535), son of Heinrich IX, lord of Greiz (d. by 1476), son of Heinrich VIII (killed 1426), son of Heinrich IV, lord of greiz (d. 1368), son of Heinrich II, Vogt of Plauen (d. 1350), Son of Heinrich I "The Russian" Vogt of Plauen (d. 1292/95), son of Heinrich I, vogt of Plauen (d. 1303), son of Heinrich IV, Vogt of Weida & Gera, (d. 1249).
So according to my count that makes at least 22 successive generations from father to son named Heinrich, and possibly more.