We analyze the structural and magnetic characteristics of (111)-oriented lutetium iron garnet (Lu3Fe5O12) films grown by molecular-beam epitaxy, for films as thin as 2.8 nm. Thickness-dependent measurements of the in- and out-of-plane ferromagnetic resonance allow us to quantify the effects of two-magnon scattering, along with the surface anisotropy and the saturation magnetization. We achieve effective damping coefficients of 11.1(9)×10-4 for 5.3 nm films and 32(3)×10-4 for 2.8 nm films, among the lowest values reported to date for any insulating ferrimagnetic sample of comparable thickness. © 2016 Author(s).