I remember someone (maybe Tamim Ansary?) writing that states that claimed to be the direct successor of the Roman Empire existed until 1922, at least in the sense that the Ottoman Empire used some titles and administrative terminology of the Byzantine Empire.
https://en.wikipedia.org/wiki/List_of_sultans_of_the_Ottoman...
> After the conquest of Constantinople in 1453 by Mehmed II, Ottoman sultans came to regard themselves as the successors of the Roman Empire, hence their occasional use of the titles caesar (قیصر qayser) of Rûm, and emperor, as well as the caliph of Islam.
A more direct name claim would be
https://en.wikipedia.org/wiki/Sultanate_of_Rum
which lasted until 1308.
A different later tradition of claiming to be the Roman emperor is the Holy Roman Empire's
https://en.wikipedia.org/wiki/Holy_Roman_Empire
which used that term after a break of several centuries (so not very continuously with the ancient Roman Empire). But Germans then claimed to be Roman emperors (in some sense) until 1806!
https://en.wikipedia.org/wiki/Francis_II,_Holy_Roman_Emperor
These aren't as continuous with the ancient Roman empire as the Byzantines, but it's still pretty astonishing to think that various monarchs were still claiming to be (in at least a theoretical legal sense) Roman emperors during the 1800s and 1900s.
