Reasoning

... the human population has not and is not growing exponentially and is unlikely to do so, says Joel Cohen, a populations researcher at the Rockefeller University in New York City. The world's population is now growing at just half the rate it was before 1965. Today there are an estimated 7.3 billion people, and that is projected to reach 9.7 billion by 2050. Yet beliefs that the rate of population growth will lead to some doomsday scenario have been continually perpetuated. ...

The world's population also has enough to eat. According to the Food and Agriculture Organization of the United Nations, the rate of global food production outstrips the growth of the population. People grow enough calories in cereals alone to feed between 10 billion and 12 billion people. Yet hunger and malnutrition persist worldwide. This is because about 55% of the food grown is divided between feeding cattle, making fuel and other materials or going to waste, says Cohen. And what remains is not evenly distributed - the rich have plenty, the poor have little. Likewise, water is not scarce on a global scale, even though 1.2 billion people live in areas where it is.

"Overpopulation is really not overpopulation. It's a question about poverty", says Nicholas Eberstadt, a demographer at the American Enterprise Institute, a conservative think tank based in Washington DC. Yet instead of examining why poverty exists and how to sustainably support a growing population, he says, social scientists and biologists talk past each other, debating definitions and causes of overpopulation.

> Is there an estimated maximum possible
> population of the Earth ?

No. And there fundamentally cannot be.

Humans are unusual creatures because our behavior, our "ecological niche", and our use of resources are not determined purely genetically as it tends to be for other animals. We are fundamentally a technological species. That doesn't just mean we use tools, as some people have come to take the meaning, it means that we are capable of adapting, learning, and teaching. Which means that we are capable of passing on how to build and use a specific tool, of course, but it's much more than that. It also means we can pass on non-corporeal tools, tools of the mind and of behavior, tools such as language, culture, ethics, art, etc.

What this means for the human species is that the coupling between "natural environmental limits" and the limits of population is effectively broken. If you take, say, a group of deer you can draw strong connections between their environment and the degree to which it can support a population of a given size, if you get too many deer there just won't be enough food, etc.

Humans, however, do not have fixed behavior. More so, we are capable not just of consuming resources but of producing, or causing the production of, resources. Humans have the ability to change their diet, change how they live, where they live, etc. And we produce food, produce energy, produce raw materials, etc. For a given level of technological / industrial capability and a given set of human behaviors you could make a claim about the carrying capacity of the Earth, however none of those things is constant.

Take food, for example. The amount and type of meat that people eat greatly impacts the amount of resources it takes to supply food for that person. Also, developments such as more efficient means of farming, new strains of crops, and artificial fertilizer have enabled massive increases in the efficiency and effectiveness of food production. If there were some particular major limiting factor on some aspect of human living then there would be pressure, due to economics etc, to change behaviors and avoid that limit. For example, if we could no longer supply much meat to people then folks would switch to more vegetarian diets. If we ran into a limit in production of copper then people would turn to other materials like aluminum or iron or plastic for many of the uses of copper. And if we ran into a problem with producing fresh drinking water people would start changing their water usage patterns, we'd stop washing our cars, watering our lawns, and flushing our toilets with drinking water, for example. And so forth.

And that's the fundamental problem of estimating the human carrying capacity of the Earth. You're dealing with two major moving targets (behavior/lifestyle and industrial/agricultural capability) each of which breaks down into innumerable smaller areas that are all of enormous complexity. Any attempt at a holistic analysis is going to be a gross oversimplification one way or another.

Yes ! Archaeologists and human paleoecologists haven't considered population limits (or "carrying capacity", to use the technical term) to be something directly measurable for decades. The best you can do is construct mathematical models, and these oversimplify the factors that affect it to the point where the information being produced is effectively meaningless.

For a more in-depth explanation of the problems associated with the Malthusian/Leslie White approach to population limits (or "carrying capacity" to use the technical term), see this article by archaeologist Brian Hayden.

It's not layman speculation. This is a standard response given by archaeologists (who study changes in human environment interactions over thousand year timescales, and are thus directly relevant to this question). Here's another article backing that position.

The estimates of Earth's carrying capacity range from under 1 billion to more than 1000 billion persons. Not only is there an enormous range of values, but there is no tendency of the values to converge over time; indeed, the estimates made since 1950 exhibit greater variability than those made earlier.