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Yes, those info were also new to me before knowing about SB, which mentions more than 20 avtaars of Vishnu thr't the whole text. As the 'Rules' post does not have any point mentioning whether to include them (relatively lesser known Avtaars of Vishnu from SB)  in our game Thread or not, so I decided to include them here.  

Yes, there are actually 22 avatars as per Srimad bhagavatam. Dhanvantri is one of them- as is Mohini.  The avatars are listed in the First Skanda, chapter 3.

Regarding the progeny of Vyasa- 

Wilst Dritarashtra, Pandu , Vidura and Shukadev are all biological sons of Vyasa, only the last can be considered his  actual son . Though the first three were also begot by him, they automatically came under the 'Kuru' gotra- not Vyasa's gotra. 

Let me try to explain this (as much as I have understood...). Dritarashtra and Pandu were fathered by Vyasa- no doubt, but if they had to perform their father's tarpan who would it be? Vichitraveerya's  and even Chitrangada's but certainly not Vyasa's . Why? Because of the gotra (family lineage) into which they are born into. 

For instance, when Dritarashtra and Pandu's Upanayanam would have been performed, whose names they would have taken as their father and grandfather? Vichitraveerya and Shantanu's - not Krishna Dwaipayana and Parashara. These names would have been taken by Shukhadev.

That's why I feel, Vyasa cannot be considered Kunti's father in law . Father of Drt and Pandu...maybe but not father in law of Kunti / Gandhari / Mrs Vidura. 

Emperor Pratip's grandfather.  Dushyant????

 There not be any speculations. Srimad Bhagavatham lists all the 22. Narad is there, as is Kapila muni. Nara / Narayana are also included, but whether one or  two I am not sure. Dattatreya is also counted...

What Valmiki Ramayan has to say about LBS, I am not sure...

Coming back to your question, Kunti herself is a Vrishi / Shoora by birth so why should she not be considered as a descendant of Devyani / Yayati ?

I agree w/ Varaali in that although Pandu can be considered Vyasa's son, Kunti can't be considered his DIL for the reasons she described.  Like the Upanayan and all that, even a marriage would have made Kunti the kulvadhu of Vichitravirya/Ambalika, not Vyasa.

Anyway, my answer to your question here is Ajmidha

You went from Janamejaya - Parikshit - 1 generation

Parikshit - Arjun - 2 generations

Arjun - Shantanu - 3 generations

So the answer would be 4 generations above Shantanu.  Shantanu's father was Jahnu, whose father was Kuru, whose father was Ruksha, whose father was Ajmidha.

Am I correct in either line of reasoning, or actual answer?  If I'm right in the former but wrong in the latter, then who do you have 4 generations above Shantanu?

Hasti was the father of Ajmidha, and 4 generations after Bharata.  I don't think this question would include Bharat, since it went just by Janamejaya's ancestrial lineage, which is why Shivang put Arjun instead of Yudhisthir in the chain.  But it may have been Shivang's answer, assuming that he may not have expected anyone to know about Ajmidha.

Wasn't Shantanu's father  Emperor Pratip?