The cytoskeleton (CSK) is a crowded network of structural proteins that stabilizes cell shape and drives cell motions. Recent studies on the dynamics of the CSK have established that a wide variety of cell types exhibit rheology in which responses are not tied to any particular relaxation times and are thus scale-free. Scale-free rheology is often found in a class of materials called soft glasses, but not all materials expressing scale-free rheology are glassy (see plastics, wood, concrete or some metals for example). As such, the extent to which dynamics of the CSK might be regarded as glassy remained an open question. Here we report both forced and spontaneous motions of microbeads tightly bound to the CSK of human muscle cells. Large oscillatory shear fluidized the CSK matrix, which was followed by slow scale-free recovery of rheological properties (aging). Spontaneous bead motions were subdiffusive at short times but superdiffusive at longer times; intermittent motions reflecting nanoscale CSK rearrangements depended on both the approach to kinetic arrest and energy release due to ATP hydrolysis. Aging, intermittency, and approach to kinetic arrest establish a striking analogy between the behaviour of the living CSK and that of inert non-equilibrium systems, including soft glasses, but with important differences that are highly ATP-dependent. These mesoscale dynamics link integrative CSK functions to underlying molecular events, and represent an important intersection of topical issues in condensed matter physics and systems biology.