Rome fits the bill, depending on how you define "exist." FYI, I'm no expert in Ancient Rome (I wrote my thesis on post-WWII Austria), so apologies for any inaccuracies - it's the first example that came to mind.

Rome started out as a Kingdom, lasting from 753 BC, which is the traditional "founding" of the city of Rome by Romulus and Remus. That's more mythology than history though. In 509 BC, Rome became a republic after the kings were expelled from the city (219-221). By 264 BC, Rome controlled most of the Italian peninsula, extending north beyond Trasimene, not yet bordering the Southern Alps. By 100 BC, the Roman Republic controlled most of the lands comprising the northern coast of the Mediterranean, including the southern coast of Spain, all of Greece, most of Turkey by about 75 BC, and parts of Palestine by 58 BC (233). Gallia was conquered by 61 BC. Of course, this is extremely condensed.

After the downfall of the Republic, Rome continued to expand to conquer England, Northern Africa, Palestine (Judea), and Egypt (243). In 330 AD, Emperor Constantine "split" the Empire, founding "a city to rival Rome," in Byzantium. Because the Roman Empire had grown "too big for a unified political structure," Constantinople ruled over the Eastern Empire and Rome over the West. Skip forward to the fall of the Western Empire (278).

476 is the year historians give to the "end" of the Roman Empire in the West, but it was more of a gradual thing. What's important is that by 500 AD, "the boundaries of the eastern empire were still much what they had been under Constantine, and his successors were still holding their own against the Persians, [and] the last western emperor had been deposed and his insignia sent to Constantinople by a barbarian king who claimed to rule as the eastern emperor's representative in the West" (279).

The Eastern Roman Empire, known as the Byzantine Empire, fell in 1204 Crusaders, existed in exile, and was then reinstated in 1264 (349-349) Ultimately, Constantinople fell to the Ottomans in 1453 (374).

So one could claim that the Roman Empire lasted until 1453, but the Byzantine Empire had grown into something totally different. I'd equate it to biological evolution. Etruscan civilization becomes the Kingdom of Rome, which became the Roman Republic, which became the Roman Empire, which split into the East and the West. After the West collapsed, the Eastern Empire lived on, evolving, and further distancing itself in almost every way from what the Western Empire was.

There was another thread asking why France is considered older than the US, even though the Fifth Republic isn't even sixty years old. The answer given was: France has existed in one form or another since 843 AD, the United States "only" since 1776. I'm kind of using this logic here.

So, in a way, the Roman Empire endured for a little less a thousand years after the the sack of Rome in 455.

Source: The Penguin History of the World by J.M. Roberts.

It might not be the most comprehensive source, but most of my books are in storage, so it'll work. Most of this information can also be found by browsing Wikipedia.

It depends on what you consider a country, and how you define their original boundaries.

The Roman empire survived in the east for nearly a thousand years after Italy had been lost (although parts of it were very briefly recaptured).

Taiwan, or the Republic of China, was essentially overthrown and "exiled" to Taiwan while losing their grip on the mainland during a civil war. This one might count depending on how you view it. China did own Taiwan before this happened, but it was far from what I would consider a core territory of China. Technically, at the founding of the RoC, Taiwan was owned by the Japanese, so I guess it should count.

France is actually an example of such country as the Frankish homeland was in what is now Belgium, the Netherlands and Germany.

It's a question if the Frankish homeland qualifies as "a country" though...

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again. Alternatively, questions of this type can be directed to more appropriate subreddits, such as /r/history or /r/askhistory.