I am aware of the Eastern Empire being Roman for a long time after the fall of the Western Empire, but it's notable how parallels seem to have been drawn in the HRE, Third Reich, and Russia with Rome. I've also read about claims to legitimate succession in Trebizond and other splinter kingdoms. So how many nations in all have claimed either direct or "successive" descent from the authority of Rome?
As far as I know, this is the list and short explanation of why.
The Roman Catholic Church saw itself as the immediate heir to the empire after it fell.
The Church made the Frankish King Clovis I the Holy Roman Emperor, which made them the successor to the Roman Empire.
When Charlemagne died and divided his empire, he made the Germanic portion the Holy Roman Empire. this led to the elector princes chosing the HRE.
When Napoleon ascended as Emperor of France, he claimed that since France was part of Charlemagne's, he was now HRE.
At the same time, the Hapsburg Empire ruled Austria-Hungary which was the HRE from number 3. In 1814, with the abdication of Napoleon as Emperor, the Hapsburgs took the title again.
After the fall of Constantinople, the Ottomans considered themselves the true heirs of the Romans since the conquered the Eastern Roman Empire.
The Russians claimed the title until 1917 as well since the Eastern Orthodox Church was under their auspices.
At various times, the British Empire claimed to be the spiritual successor to Rome.
Finally, the Third Reich said they were as well, though no one took them seriously.
Basically, everyone but Italy has claimed the title since the fall of the actual Roman Empire.