From the diminishing sound of a plucked guitar string to the quick fading of ripples on water, wave attenuation in lossy media is a ubiquitous phenomenon. Here, we unveil a hidden connection between this familiar effect and a time-domain non-Hermitian topology. We show that a broad class of velocity-dependent dissipation, such as everyday air damping, can essentially break temporal reciprocity by introducing an imaginary gauge field along the time axis. This gauge field opens a point gap in momentum bands and stimulates a temporal analogue of the non-Hermitian skin effect, underpinning the topological origin of loss-induced wave attenuation. Despite its simplicity and ubiquity, this temporal nonreciprocity serves as a powerful knob for sculpting both the steady-state and dynamic responses of emergent time crystals, from braiding momentum bands in the frequency Brillouin zone to reshaping interface states in the temporal domain. Moreover, the inherent causality of time grants unprecedented flexibility in controlling boundary conditions and system size—features prohibited in conventional spatial models—as we experimentally demonstrate using a simple mechanical platform.