To enable suppression of two-magnon scattering (TMS) in nanometer-thick Co (ultrathin Co) layers and realize low-magnon damping in such layers, the magnon damping in ultrathin Co layers grown on various nonmagnetic seed layers with different surface magnetic anisotropy (SMA) energies are investigated. We verify the significantly weak magnon damping realized by varying the seeding layer species used. Although TMS is enhanced in ultrathin Co on Cu and Al seeding layers, the insertion of a Ti seeding layer below the ultrathin Co greatly suppresses the TMS, which is attributed to suppression of the SMA at the interface between Co and Ti. The Gilbert damping constant of the ultrathin Co layer on Ti (3 nm), 0.020, is comparable to the value for bulk Co, although the Co layer thickness here is only 2 nm. Realization of such weak magnon damping can open the door to tunable magnon excitation, thus enabling coupling of magnons with other quanta such as photons, given that the magnetization of ultrathin ferromagnets can be tuned using an external electric field.