I was thinking recently about the possibility of the Sumerian language surviving into modern times. Obviously a highly unlikely scenario (and one that depends quite a bit on when exactly you think it died out) in a lot of ways. Though I tend to think that people are often too fatalistic about history, thinking "it happened so it must have bean almost inevitable".
Obviously whatever did actually happen in any given example is likely to be whatever was most likely to happen (although unlikely things do happen occasionally as well) or at least one of the more probable things that could have happened so there is some merit to that kind of thinking but nonetheless I think that there are a few reasonably realistic ways that Sumerian could have survived into modern times.
Not as the majority language of the region probably given the sheer amount of language replacement that happened over the historical time-span. Something like Sumerian > Akkadian > Amorite ( > Babylonian Akkadian) > Kassite ( > Assyrian Akkadian) > Chaldean > Aramaean > (Some back and forth including Greek and Persian sort of) > Arabic. So given that history it seems unlikely that Sumerian would manage to remain dominant over that entire time-span. But that said if it could just manage to survive a bit longer perhaps it could then go on to survive as a minority language somewhere in the region.
I see two fairly straight-forward ways that this could have happened. The first is rather obvious. Assume that Sargon the Great never existed, or failed in his attempt to conquer Mesopotamia. Then the first securely attested empire could have been a Sumerian one. Honestly from the stories about Sargon I get the impression that his success was quite the stroke of luck and strategic genius and actually very unlikely.
This would then have increased the prestige of Sumerian and possibly even spread it further outside of the classical Sumerian heartland.
The most probable thing to happen after that is presumably that the Sumerian Empire falls to the Gutians like the Akkadian one eventually did, but of course the sky is the limit in theory.
The second way that it could have perhaps survived is if the Ur-III kings managed to do something about the salt buildup in the soil. Which as I understand it is a problem that was solved (or at least partially solved) by Mesopotamians over the next five centuries or so. And the solutions don't seem that counter intuitive and they had the records to prove what was happening so this doesn't seem too unlikely if they had choosen to focus on this problem (though political realities might have made this difficult to do in practice and it may require some luck or more time than they really had). Couple that with a longer reign for Shulgis succesor, or perhaps a more competent one, and I think that the UR-III dynasty could have survived over the Issin-Larsa period and perhaps even replace the Old Babylonian Empire.