Familiar chinese saying:
富不过三代 or Wealth does not pass three generations.
This aphorism relates not just to monetary wealth, but also towards family fortunes in their entirety –power and influence– which naturally includes the rise and decline of monarchs and dynasties.
Erroneous apocryphal aphorismus application aside (there are actually many instances of family fortunes of wealth, monarchial power remaining more or less distinct and pure beyond 3-gen), three generations does seem to be a rather apt interval for accumulated fortunes, of any kind, to start dissipating.
[A natural occurrence after all: assuming an even steady-state reproduction ratio of 2.1 for the founding Patriarch, by the 3rd-gen, there would already be 4-going-on-8 descendants, leading to a natural dissipation of accumulated resources. (and most alpha-male progenitors would be highly polygamous and have reproduction factors way excess of 2.1)
But there may be an interesting counterpoint to this in some cultures which concentrates power and resources in only one person within each successive generation (an almost ‘disinheriting’ of all others) in order to preserve, concentrate and continue the progenital line.
(the continuing success and relative longevity of post-Meiji modern japanese corporations have some of the flavour of this)
One of the more celebrated instance of 富不过三代 or Wealth does not pass three generations is the Mongol Empire itself, with Genghis Khan as the progenitor.
While significant and large parts of the Mongol Empire lasted quite many generations past the third, the unified Mongol Empire itself irreparably splintered less than 30 years after the death of Genghis Khan, with the mutedly opposed passing of the Khanate-ship from the House of Ogodei (the anointed successor to Genghis) to the House of Tolui into the hands of Tolui’s son, Mongke (3rd-gen) in 1251. The Toluid reign of Great Khans, first through Mongke and subsequently through his brother Kublai/Qubilai was initially tolerated and towards the end, openly challenged by their cousins, the almost equally-strong Batu khan (son of Jochi) of the Golden Horde (north-west Central Asia, Russia) and the Chaghatayid khans (sons of Chaghatai) of Central Asia.
But Mongke held onto his reign, with the support of his brothers and their infamous strong large armies: Kublai (campaigning still in China for the remaining parts of the Chinese Song Dynasty) and Hulegu (his terrifying almost unstoppable sweep of Persia and Arabia, almost wiping off the Muslim world from the face of the earth; until he met the Mamluks of Egyt anyway).
After Mongke died and Kublai became Great Khan in 1260, brothers eventually turned on brothers, with Kublai’s younger brother Ariq Boke’s vying and grabbing for the throne marking the start of the Mongol civil war, in all parts of the sprawling empire.
The largest land empire in history, commanding unimaginable wealth and power, could not last pass three generations and 30-odd years. Genghis Khan himself likely saw this coming and was probably familiar with the following tale from his ancestors:
Their mother Alan Qo’a knew what they had been saying to each other behind her back.
One day in spring, while she was cooking some dried lamb, she had her five sons Belgünütei, Bügünütei, Buqu Qatagai, Buqatu Salji and Bodon_ar Mungqaq sit in a row. She gave an arrow-shaft to each of them and said, ‘Break it!’ One by one they immediately broke the single arrow-shafts and threw them away. Then she tied five arrow-shafts into a bundle and gave it to them saying, ‘Break it!’ The five sons each took the five bound arrow-shafts in turn, but were unable to break them.
Then their mother Alan Qo’a said, ‘You, my sons Belgünütei and Bügünütei, are suspicious of me and said to each other, “These three sons that she has borne, of whom, of what clan, are they the sons? […]
Further, Alan Qo’a addressed these words of admonition to her five sons: ‘You, my five sons, were born of one womb. If, like the five arrow-shafts just now, each of you keeps to himself, then like those single arrow-shafts, anybody will easily break you. If, like the bound arrow-shafts, you remain together and of one mind, how can anyone deal with you so easily?’ Some time went by and their mother Alan Qo’a died.
—THE SECRET HISTORY OF THE MONGOLS