What will it be like when we reach our carrying capacity?
A chaos (I think)
Everything has a limit. This earth too. When we will reach our carrying capacity (I hope we will not see anytime), water, food, shelter and resources will be very limited (per capita). People will be unhappy due to hunger (or maybe due to other reasons). Wars might start to control food and/or water resources.
The Earth will be fine but will have no trees and a lot of polluted water in the ocean.
10 billion people is the limit of humans that the earth can support and higher than that will lead to a crowded world of chaos. This is because people will most likely fight for food, water and shelter because space is scarce. The current human population is around 7 billion people so it will take maybe till 2100AD for the world population to peak. But, I don't think the world population will reach 10 billion because if we are going to colonise our neighbour planet Mars, we can bring humans there and start a new home. Hope this answers your question though.