  Low-dimensional, complex systems are often characterized by logarithmically
slow dynamics. We study the generic motion of a labeled particle in an ensemble
of identical diffusing particles with hardcore interactions in a strongly
disordered, one-dimensional environment. Each particle in this single file is
trapped for a random waiting time $\tau$ with power law distribution
$\psi(\tau)\simeq\tau^{-1- \alpha}$, such that the $\tau$ values are
independent, local quantities for all particles. From scaling arguments and
simulations, we find that for the scale-free waiting time case $0<\alpha<1$,
the tracer particle dynamics is ultra-slow with a logarithmic mean square
displacement (MSD) $\langle x^2(t)\rangle\simeq(\log t)^{1/2}$. This extreme
slowing down compared to regular single file motion $\langle
x^2(t)\rangle\simeq t^{1/2}$ is due to the high likelihood that the labeled
particle keeps encountering strongly immobilized neighbors. For the case
$1<\alpha<2$ we observe the MSD scaling $\langle x^2(t)\rangle\simeq
t^{\gamma}$, where $\gamma<1/2$, while for $\alpha>2$ we recover Harris law
$\simeq t^{1/2}$.
