Don’t leave out Tamil when it’s a question of continuity of intelligibility over long periods of time. Tamil texts dated to the 1st century B.C. can still be read and understood by modern literate Tamil speakers. Sure, spoken Tamil has changed over the past two millennia, but alongside the spoken (demotic, if you will) dialects, literary Tamil is still used in everyday life. This is called diglossia.
It’s also true of Arabic. Similarly to Tamil, Arabic texts dated to the 6th century can be read by anyone who can understand Modern Standard Arabic. As with Tamil, the demotic dialects are used for ordinary conversation, even by educated people, but for writing and making speeches, the formal register is language that would be intelligible to someone visiting in a time machine from 1500 years ago. In Tamil, the intelligible time depth is known to reach at least 2200 years back.
Ancient Arabic pronunciation has been kept current all this time by study and oral recitation of the Qur’an. This has preserved the pronunciation of 1400 years ago, which is still used for Modern Standard Arabic, a living language. The pronunciation of Tamil has definitely changed over the last 2000 years, though I’m not sure how much modern standard Tamil pronunciation differs from ancient Tamil pronunciation. Probably not enough to completely destroy intelligibilty, if an ancient Tamilian were to time-travel to the present, but certainly enough to impair intelligibility.
Modern Greek did have comparable diglossia until a few years ago, when the government abolished Katharevousa. Even when they still used Katharevousa (which was an invention of 19th-century Greek nationalists, by the way), they used Modern Greek pronunciation for it. It’s often warned that we don’t know exactly how ancient Greek was pronounced. Well, we do have a pretty good idea of it with reasonable certainty, based on evidence from that time, such as transcription of Greek words into Latin, and other comparative linguistics. We do know for a fact that Greek pronunciation has changed so much over the past 2400 years that Aristotle would have at best a very, very difficult time understanding a speaker of modern Katharevousa. Classical Greek had 14 different vowels and diphthongs: [symbol]a, e, h, i, o, u, w, ai, au, ei, eu, oi, ou, ui[/symbol]. Modern Greek pronunciation has collapsed these into only 5 vowel sounds: /a/, /[symbol]e[/symbol]/, /i/, /o/, /u/. The two surviving diphthongs [symbol]au[/symbol] and [symbol]eu[/symbol] are now pronounced as vowel-consonant combinations av~af and ev~ef. I mean, this would sound like a foreign language to Aristotle’s ears.
Taking all things into consideration, one could make a strong case for Tamil being the longest-enduring living language that is still used pretty much the way it was in ancient times. So with all the linguistic caveats noted by Dopers in this thread (that the question was badly formed and is now reformulated ), I propose Tamil as the “answer” to the OP.