Research on human online activities usually assumes that total activity $T$ increases linearly with active population $P$, that is, $T\propto P^γ(γ=1)$. However, we find examples of systems where total activity grows faster than active population. Our study shows that the power law relationship $T\propto P^γ(γ>1)$ is in fact ubiquitous in online activities such as micro-blogging, news voting and photo tagging. We call the pattern "accelerating growth" and find it relates to a type of distribution that changes with system size. We show both analytically and empirically how the growth rate $γ$ associates with a scaling parameter $b$ in the size-dependent distribution. As most previous studies explain accelerating growth by power law distribution, the model of size-dependent distribution is novel and worth further exploration.