We study two nonlinear extensions of the nonlocal R\,\Box^{-2}R gravity theory. We extend this theory in two different ways suggested by conformal symmetry, either replacing \Box^{-2} with (-\Box + R/6)^{-2}, which is the operator that enters the action for a conformally-coupled scalar field, or replacing \Box^{-2} with the inverse of the Paneitz operator, which is a four-derivative operator that enters in the effective action induced by the conformal anomaly. We show that the former modification gives an interesting and viable cosmological model, with a dark energy equation of state today w_{\rm DE}\simeq -1.01, which very closely mimics \LambdaCDM and evolves asymptotically into a de Sitter solution. The model based on the Paneitz operator seems instead excluded by the comparison with observations. We also review some issues about the causality of nonlocal theories, and we point out that these nonlocal models can be modified so to nicely interpolate between Starobinski inflation in the primordial universe and accelerated expansion in the recent epoch.
