There is an amazing amount of research being put in place toward aging. Of interest is an article on the difference in cell proteins of young vs old animals (http://www.cell.com/fulltext/S2405-4712%2815%2900110-6). Apparently there is only a 10% variation in the proteins that are produced in old vs young animals. On top of that, the variation is not consistent across all organs. This makes sense as some tissue regenerates rapidly (e.g., liver, stomach lining, skin) while other tissue does not (e.g, neurons).
Ok. So, our organs will go bad at different rates. Livers and hearts should be right around the corner. See the likes of Revivicor (http://www.revivicor.com/) which is working diligently toward the genetic humanification of pig organs via genetic manipulation. But neurons? There in lies the rub and the need for additional research in neuro-degenerative diseases such as Parkinson’s, Alzheimer’s, frontal-lobe dementia and the like. I cannot think of anyone who wishes to live forever in a fit mindless body waiting to have their diaper changed.
So, what happens if we fix these maladies? Well, there is currently a balance between the growth and death rate of the human population that is somewhere below 2%. This growth rate is not uniformly distributed across the planet’s geography. (https://www.cia.gov/library/publications/the-world-factbook/rankorder/2002rank.html). Along with this is an uneven distribution of arable land across the planet (https://en.wikipedia.org/wiki/Arable_land).
If we gave 0.25 Acres of land toward food per person, and there are 13958000 acres of arable land, then 13,796,387,658 people could be fed. Of course, this assumes good weather, insects at bay, and healthy crops all around. If there are 7.4 billion people currently on the planet (https://en.wikipedia.org/wiki/World_population) then in theory we could increase the population by 160%.
Looking at the clever graphs provided by the United Nations (https://esa.un.org/unpd/wpp/Graphs/Probabilistic/POP/TOT/) we are probably good beyond 2100 iff all goes well. Looking at a 90% projection (~12.4B people) says we have until about 2080 (looking at the linear projection from the observed line) before we hit our die-off. Stuart McMillen did a fine job of illustrating the situation (http://www.stuartmcmillen.com/comics_en/st-matthew-island/).
Time for another cuppa.