High wave-vector magnon excitations in ferromagnetic Fe monolayer and double-layer grown on W(110) are investigated using spin-polarized electron energy loss spectroscopy. The magnon dispersion relation is obtained up to the Brillouin zone boundary. A direct comparison among different systems shows that the magnons in the Fe monolayer are extremely soft and are even softer than the acoustic surface mode of Fe(110). By measuring the spectra in both energy loss and gain regions on a double-layer Fe film at room temperature and by reversing the sample magnetization, it is demonstrated that the magnon dispersion is asymmetric with respect to the sign of the wave-vector. The asymmetric dispersion relation is attributed to the degeneracy breaking of the magnons due to the presence of the Dzyaloshinskii-Moriya interaction.