Here, we report the photoconducting response of field-effect transistors based on three atomic layers of chemical vapor transport grown WSe$_2$ crystals mechanically exfoliated onto SiO$_2$. We find that tri-layered WSe$_2$ field-effect transistors, built with the simplest possible architecture, can display high hole mobilities ranging from 350 cm$^2$/Vs at room temperature (saturating at a value of ~500 cm$^2$/Vs below 50 K) displaying a strong photocurrent response which leads to exceptionally high photo responsivities up to 7 A/W under white light illumination of the entire channel for power densities p < 10$^2$ W/m$^2$. Under a fixed wavelength of $lambda$ = 532 nm and a laser spot size smaller than the conducting channel area we extract photo responsitivities approaching 100 mA/W with concomitantly high external quantum efficiencies up to ~ 40 % at room temperature. These values surpass values recently reported from more complex architectures, such as graphene and transition metal dichalcogenides based heterostructures. Also, tri-layered WSe$_2$ photo-transistors display photo response times in the order of 10 microseconds. Our results indicate that the addition of a few atomic layers considerably decreases the photo response times, probably by minimizing the interaction with the substrates, while maintaining a very high photo-responsivity.
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