The conformal loop ensemble CLEκ is a canonical random ensemble of loops which is conformally invariant in law, and whose loops locally look like Schramm-Loewner evolution with parameter κ. It describes the scaling limits of the Ising model, percolation, and other models. When 8/3 < κ ≤ 4 the loops are simple curves. In this regime we compute the three-point nesting statistic...