Disc truncations are the closest feature to an edge that galaxies have, but the nature of this phenomenon is not yet understood. In this paper we explore the truncations in two nearby (D ˜15 Mpc) Milky Way-like galaxies: NGC 4565 and NGC 5907. We cover a wide wavelength range from the NUV and optical to 3.6 {μ m}. We find that the radius of the truncation (26 ± 0.5 kpc) is independent of wavelength. Surprisingly, we identify (at all wavelengths) the truncation at altitudes as high as 3 kpc above the mid-plane, which implies that the thin disc in those outer regions has a width of at least this value. We find the characteristic U-shaped radial colour profile associated with a star formation threshold at the location of the truncation. Further supporting such an origin, the stellar mass density at the position of the truncation is ˜1-2 M⊙ pc-2, in good agreement with the critical gas density for transforming gas into stars. Beyond the truncation, the stellar mass in the mid-plane of the disc drops to just 0.1-0.2 per cent of the total stellar mass of the galaxies. The detection of the truncation at high altitude in combination with the U shape of the radial colour profile allows us to establish, for the first time, an upper limit to the present-day growth rate of galactic discs. We find that if the discs of these galaxies are growing inside-out, their growth rate is less than 0.6-1 kpc Gyr-1.