An upper bound can be generated somewhat easily. The sun continues to grow larger and larger, and one day it will become so large, in what is called the red giant stage, that it will consume the Earth. The Earth will have become thermally uninhabitable long before then. This means that humanity, if it wishes to continue to survive, must move somewhere away from Earth before Earth's thermal death. This event is billions of years in the future, and it is much more likely that some event will happen before then to threaten the existence of humanity.
The title question might have several answers, depending on the lifestyle situation of the people. Since it began, human society has been progressing through different forms, which were controlled by the development of technology. The earliest days of human society were back in the Stone Age, many millennia ago. Then someone invented a process for producing copper and then bronze, which allowed some changes to society. Following these developments, iron was discovered, probably simultaneously with charcoal, which permits smelting at a higher temperature. Agriculture was started somewhere in these early ages, and animals became domesticated, both as food sources and as work animals. Wind began to be used, with sails for water transportation and mills for grinding grain. Flowing water was also harnessed for mills.
All this set the stage for the discovery and utilization of fossil fuel energy, first coal, then oil and finally natural gas. These fuels led to a large increase in excess energy, and science and technology began a very serious period of development. Technology led to the utilization of steam power, and later electricity was invented. This led to a plethora of inventions, including electronics and microelectronics, which led to computers. The stream of inventions continues to flow strongly and constantly. This involves the mining of a great many different ores around the world and their transportation.
We might ask: How long can humanity survive at the agricultural level, where more concentrated forms of energy have disappeared? We might also ask: How long can humanity continue to exist at the high standard of living we now possess? But this ignores the obvious fact that society will continue to develop new scientific breakthroughs and technological discoveries. The rate of change is astounding, compared to earlier eras; it is unlikely that society only one century from now will be very similar to what it is now. This problem may seem unsolvable, but it is not. It is fortunate for our discussion that science is not infinite, but instead there are only a finite amount of discoveries that can be made. As more are completed, diminishing returns sets in, until some asymptotic value is reached. Science doesn't stop entirely, but since almost everything will have already become known, there are only some smaller details that need to be figured out. Thus, we can also ask, how long can humanity survive at the era of asymptotic technology?
Society might be capable of destroying itself, so we need to ask these three questions under the presumption that humanity has started thinking about its long term future and can make the choices necessary to avoid damaging its future prospects or even committing social suicide, as with a nuclear war or horribly malevolent virus. Now the questions appear to be more nicely framed: given that humanity adopts the goal of lasting as long as possible, how long would this be for us, at an agricultural level, at the current industrial and electronic level, and at the asymptotic level?
What exactly does it mean to make choices to promote the long-term survival of humanity? It is necessary to come up with a list of events that might put an end to us and then see what mankind might do to avoid them. At the agricultural level, having the soil become depleted and gradually produce less and less food every year is one possible problem. This problem is quite noticeable and could be coped with, over the course of many decades, by the means already devised for preventing soil death. Growing fewer crops is the simplest solution, and growing non-edible restorative crops certain years is another. But there must be enough food to feed the whole population, so the essential answer is not to allow the population to exceed the number that can be fed using sustainable practices, in the worst years of agricutural productivity. It would also be worthwhile to develop storage for a few years of food to allow humanity to get through the very worst years, which might be years in which volcanic dust so fills the upper atmosphere that sunlight is blocked and crops fail all over.
It would be most interesting if a few groups of people who like accounting and agriculture would make the calculations necessary to give us an indication of what the population and storage numbers might be. Their results would depend on some assumptions about transportation. If one region of Earth suffered from, say, serious floods, could other regions supply them for a few years by sailboat shipment of food and other recovery supplies? This might affect the calculations. As a wild guess, based on historical numbers, a half billion might be the maximum headcount for our planet in these restricted circumstances. There would also be a minimum number of people necessary to maintain all the operations of this future agricultural society.
This level 1 future society would be vulnerable to catastrophes. There are many, both geological and astronomical. The one most familiar is that of a large asteroid that somehow manages to score a direct hit on Earth. If it were large enough, all large lifeforms would die out. The asteroid might rupture the crust, leading to what is called a basalt flood, where lava from the mantle flows to the surface for thousands of years, polluting the atmosphere and killing off much plant life. Basalt floods can also occur without the intrusion of any asteroids, and there is evidence of them in many parts of the world. Another possible catastrophe is a nearby supernova, sending enough gamma rays to the surface of the Earth to sterilize everything. Less well known would be the passage near our solar system of a black hole or star, which might disrupt the stable orbit of Earth, causing it to move to a different radius or change its eccentricity, rendering it uninhabitable. A massive ice age could also lead to an end to human society.
A level 2 society depends on having sufficient energy to run much machinery and computation, enabling a higher standard of living than can be accomplished with an agricultural-only society that lives on sunlight alone. Fossil fuels are what have brought us to this standard of living today, but they will be depleted soon enough, probably a few centuries more at the most. Nuclear power does seem sufficient to replace fossil fuels, perhaps with some optional additional contribution from what is called renewable power. This includes hydropower, wind farms and solar panel acres. It is not clear what will be the constraining resource in such a society. Perhaps it will be thorium, perhaps lithium, perhaps rare earths, perhaps something entirely different. These calculations cannot be made at the present, as we do not know exactly how such a society would be organized. The profligate use of resources would certainly have come to a halt, but there would necessarily be some resources used every year, and some one of them would run out sooner or later.
Such a level 2 society would also be subject to extermination by the same list of catastrophes that the level 1 society would, but there is a saving grace. It may be possible to establish self-sustaining colonies on other planets or satellites, either within or outside of our solar system. This project is even more hazy than the project of revising society to conserve resources and be efficient at energy use. It might be possible, but there could also be some unforeseen barrier to such colonization. We are too immature scientifically to make a good call on this question. However, the length of time that humanity can survive is very dependent on the sustainability of such an insurance colony. If the mean time between life-destroying catastrophes is a few million years, having a colony somewhere would multiply that many times over. It is not even necessary that the colony prepare to reconstitute life on Earth after the catastrophe, as long as it can continue to survive and someday establish an insurance colony of its own. A recent book, “Looking Back from Luna”, is built on the hypothetical possibility that a sustainable colony can be established on the moon, and one theme of the book is the discussion of these colonists about what they might be able to do were a catastrophe to strike Earth, but not them.
The level 3 society takes some real imagination to contemplate. This society would have gone through several future technological revolutions. These would likely include a genetics revolution, after which genetics would be as controllable as software and would become industrialized like software. It would also likely include a neurological revolution, where we learn how to maximize both the non-verbal and verbal learning of humans, and thereby increase their resulting intellectual capability. This would have education and psychological developments as fundamental components. There would also likely be an organizational revolution, where politics is replaced by some form of governance that looks forward to the long-term prospects of human society.
Even though there are many, many options for organizing such a society, and few of them could be determined by us now, they would all still be subject to the similar limitations on energy and mineral resource use, just as a level 2 society would be. This means they would have similar expected endurance as a very-high-tech society as level 2 would have, within a factor of ten most probably, and would require the same option of a self-sustaining colony as insurance against catastrophes.
So the answer to the title question is much the same for all levels of society, and it is the mean time before a catastrophe hits. Is this a million years? Can Earth's resources be stretched to last this long? We are not even capable of making this estimate except as a simple guess. It is clear, however, that for levels 2 and 3, if the population becomes serious about having humanity last for a tremendously long time, measured in millions of years, figuring out how to establish colonies on other planets, which would be self-sustaining for long periods, is perhaps the most important project we can undertake.
