If your civilization expands at a cubic rate through the universe
You're picturing the far-future civilization as a ball, whose boundary is expanding at a constant rate. But I think a more plausible picture is a spherical shell. The resources at the center of the ball will be used up, and it will only be cost-effective to transport resources from the boundary inwards to a certain distance. If the dead inner boundary expands at the same rate as the live outer boundary, we'll be experiencing quadratic, not cubic growth.
You know, you're right. I will change my reply accordingly henceforth - linear population growth, linear increase in energy usage / computing power, and quadratic increase in (nonenergetically stored) memories.
I might need a better title (It has now been updated), but here goes, anyway:
I've been considering this for a while now. Suppose we reach a point where we can live for centuries, maybe even millenia, then how do we balance? Even assuming we're as efficient as possible, there's a limit for how much resources we can have, meaning an artificial limit at the amount of people that could exist at any given moment even if we explore what we can of the galaxy and use any avaliable resource. There would have to be roughly the same rate of births and deaths in a stable population.
How would this be achieved? Somehow limiting lifespan, or children, assuming it's available to a majority? Or would this lead to a genespliced, technologically augmented and essentially immortal elite that the poor, unaugmented ones would have no chance of measuring up to? I'm sorry if this has already been considered, I'm very uneducated on the topic. If it has, could someone maybe link an analysis of the topic of lifespans and the like?