From an archaeological standpoint, every city is built on top of an older city. Even if there is no point in which the city is uninhabited layers still form because of an accumulation of individual destruction events. If, say, a house burns down the area will be swept up, piled over and a new structure will be built on top of it. Over a couple of centuries, this process will happen to virtually every place in the city, and so you have a bit of a ship of Theseus in that there is an old city under the new city even though you can't really tell when that happened.
To more directly answer your question, Damascus is the conventional answer, but that should be seen as exactly that: conventional rather than backed by thorough archaeological research.
Hasankeyf in southeast Turkey is generally considered one of the oldest continually inhabited settlements in the world, dating back at least 10,000 years with a small population today, including some people that still live in the ancient cave dwellings. Unfortunately much of the town, including ancient ruins, will likely be submerged once a massive dam project on the Tigris is completed. I was lucky enough to visit last spring, if you're interested in seeing photos of what it looks like today
Because this has been hinted at in other posts, but not explicitly stated, I am going to make an important point here: the answer to this question is going to vary considerably. As /u/Tiako already pointed out, no cities exist that are the same city they were thousands of years ago.
One of the biggest things to consider is that many of the oldest inhabited sites, like Damascus, can trace their origins back many thousands of years, but for most of their history, they were not cities. A small town on an important high point is usually the most common site for these types of settlements, since unlike rivers, mountains and hills generally don't change that much over the timespan we are talking about. But, if we take modern Damascus for example: the modern city as as much in common with the early settlement as New York City has in common with early aboriginal settlements on Manhattan. That is to say, the scale and peoples of the modern city have no real connection other than geographic with those early settlers from several thousand years ago.
Another good example is Athens. We might think, given that it is now the capital of Greece, that Athens's importance has continued since antiquity and the modern city can lay claim to much of the same history. The fact is that this simply isn't true. For most of the medieval period, particularly after 1000AD, Athens slipped into the status of a provincial backwater. The reason it is important today is because it was selected and built as a new capital for Greece, owing in large part to the large role Athens has in western consciousness. Sure, the the Acropolis survives, but the vast majority of people who live there can't trace their family history back to Attica more than a couple of generations at most. While they are still Greek in culture, many of them range from the North and particularly the former Ottoman territories now part of other countries. Is this modern Athens a completely different city? Is it the same city because of the name?
There is no official way to answer this question in a satisfactory way, and certainly no way to answer it completely. There are many settlements, some very small, that likely have histories that are just as long as Damascus or Jericho; perhaps even longer! However, the evidence may not be there, or it may remain undiscovered because that settlement is minor. Damascus gets a lot of attention because it has been a relatively important city for the past two or three millennia, but prior to that it was not a site of importance.