Tyrion Lannister: winner of the game of thrones.
If Dany had had an heir, they could've potentially claimed the throne. But, she didn't and her only other living relative was Jon. Which brings us to our next point...
Jon, as far as we know, was Dany's last blood relative, so there's no one else in the Targaryen line to take the throne after Dany.
According to laws and tradition in Westeros, the next in line to the throne would be the previous crown's next male relative.
'Cause he's gonna be busy looking for Drogon, apparently.