We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a)>0 after initiation of the process at t=0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age t(a) of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.