The last Roman successor state in the West (the Kingdom of Gwynedd: its boundaries always shifted about, but it covered most of North Wales most of the time) fell in 1283 and the last Roman successor state in the East (the Duchy of the Archipelago: essentially the Cyclades) fell in 1579. Some people would not count the Duchy of the Archipelago as it was created by Venetian adventurers/pirates during the general chaos following the Fourth Crusade, in which case the answer would be the Principality of Theodoro (a tiny Gothic-speaking slice of southern Crimea that was nominally attached to the equally tiny Empire of Trebizond) which fell in 1475. None of these places were fully sovereign all of the time by a modern understanding of the term, but that's not really important: what matters is that they were all clearly defined polities. Anyway, there's nothing afterwards.
HRE could be considered a revival of the Western Roman Empire, just as the Russian Federation could be considered a revival of the Russian Empire, in my eyes.
I was going to make a comment joking about how the answer is clearly Germany because of the HRE -> Confederation of the Rhine -> North German Confederation -> Germany succession, but it’s less funny when there’s people in thread actually claiming that the HRE was a successor of the Roman Empire.
It used the Roman language, practiced the Roman religion, and was considered to be Roman by the people who lived in and around it. That doesn’t mean it has to be a successor state, but I could see an interpretation of the terms “successor state” and of historical analysis at the time to determine that maybe it was.
All those statements have big asterisks attached though.
The vernacular was not a Roman language in most of the HRE. Most of the common people did not speak Latin-derived languages, and essentially none of them actually knew Latin. The Church used it, and laws and literature were typically written in Latin, but that was the case in all of Western Europe at that time. Writing in the vernacular was pretty uncommon until the early modern period, if I remember correctly.
The did practice Catholicism, but again, that wasn't unique to them. At the founding of the HRE they weren’t the only Catholic nation in Western Europe, let alone Europe as a whole. You also have to consider that by the end of the HRE’s existence a significant portion of it was Protestant.
When it comes to recognition, again, there’s a lot to be debated. Their claim was far from universally accepted even at the time, but that’s an issue I won’t pretend to know much about (maybe one of our more historically-knowledgeable posters could weigh in on this one).
I suppose one could interpret the term “successor state” widely enough to include the HRE as a successor state to the Roman Empire, but at that point the term is so general that it’s almost useless. They’re two very different states at the end of the day. I think the whole question is a silly one. Rome has no successor. It’s gone. A lot of their innovations and culture live on in various ways, but the Roman state has been completely dead for centuries.