t \geq (1−\exp(−x))/x
is equivalent to
t \geq (\exp(y)-1)/y,\quad y = -x
and so matches the requested atom at How to express this convex function in CVX. We still haven’t found a DCP formulation in the currently supported set of cones. Out of curiosity, what are you trying to represent with (1−\exp(−x))/x (in what application does it appear)?