  We propose and experimentally measure an entropy that quantifies the volume
of correlations among qubits. The experiment is carried out on a nearly
isolated quantum system composed of a central spin coupled and initially
uncorrelated with 15 other spins. Due to the spin-spin interactions,
information flows from the central spin to the surrounding ones forming
clusters of multi-spin correlations that grow in time. We design a nuclear
magnetic resonance experiment that directly measures the amplitudes of the
multi-spin correlations and use them to compute the evolution of what we call
correlation R\'enyi entropy. This entropy keeps growing even after the
equilibration of the entanglement entropy. We also analyze how the saturation
point and the timescale for the equilibration of the correlation R\'enyi
entropy depend on the system size.
