We report the achieving of depairing current limit along $c$-axis in Fe$_{1+y}$Te$_{1-x}$Se$_x$ single crystals. A series of crystals with $T_{rm{c}}$ ranging from 8.6 K to 13.7 K (different amount of excess Fe, $y$) were fabricated into $c$-axis bridges with a square-micrometer cross-section. The critical current density, $J_{rm{c}}$, was directly estimated from the transport current-voltage measurements. The transport $J_{rm{c}}$ reaches a very large value, which is about one order of magnitude larger than the depinning $J_{rm{c}}$, but comparable to the calculated depairing $J_{rm{c}}$ $sim$ 2 $times$ 10$^6$ A/cm$^2$ at 0 K, based on the Ginzburg-Landau (GL) theory. The temperature dependence of the depairing $J_{rm{c}}$ follows the GL-theory ($propto$ (1-$T/T_{rm{c}}$)$^{3/2}$) down to $sim$ 0.83 $T_{rm{c}}$, then increases with a reduced slope at low temperatures, which can be qualitatively described by the Kupriyanov-Lukichev theory. Our study provides a new route to understand the behavior of depairing $J_{rm{c}}$ in iron-based superconductors in a wide temperature range.
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