Faced with speed as a limit to the size of the human population, the optimist will conclude that all we need to do is increase the maximum speed to continue growing the population. Nature, however, has a maximum speed that the most elaborate conceivable technology would be unable to exceed: the speed of light. If we could reach the speed of light and the speed-population relation held, we would be able to grow the population to 9.3 trillion people, or 1,400 times the current population, which represents the largest group of people who could exchange resources in a useful amount of time. This isn't to say that there can't be many such groups; they simply would not represent a single population: over time, under many different circumstances, their members would likely mutate and therefore cease to be human.
If each of these largest of groups -- or “worlds” -- occupied a sphere, and explorers continued to move outward from the Earth at the maximum speed, then the number of worlds would be roughly proportional to the volume of a sphere whose radius was equal to the explorers' distance. At the speed of light, our galaxy's 100 billion solar systems (if each is as hospitable as our own) could hold more than 1,000 times as many human relatives as there are believed to be stars in the Universe (100 billion times 100 billion, what I’ll call a “universal unit”).
As isolated as individual worlds would be, galaxies would be even more so given the large, empty expanses between them which are increasing as the Universe expands. To catch the fastest of the galaxies, I estimate that explorers would need to travel about 50,000 miles per hour, or double the current speed record in space. At that speed, a world would have 80 billion members; and our galaxy could ultimately have human relatives numbering more than 20 trillion universal units. The entire known Universe could hold more than 2 trillion-trillion universal units of relatives.
It appears counter-intuitive that increasing speed will decrease the total number of human relatives. The reason for this is that while the volume of the expanding sphere is proportional to the cube of the speed, the number of relatives is only increasing as the square root of the speed. From another perspective, fewer people are using more resources (a factor of more than 100:1 in per capita consumption between traveling at light-speed and traveling at 50,000 mph).