We experimentally demonstrate enhanced Kerr nonlinear effects in highly nonlinear As(2)Se(3) chalcogenide fiber tapered down to sub-wavelength waist diameter of 1.2 mum. Based on self phase modulation measurements, we infer an enhanced nonlinearity of 68 W(-1)m(-1). This is 62,000 times larger than in standard silica singlemode fiber, owing to the 500 times larger n(2) and almost 125 times smaller effective mode area. We also consider the potential to exploit the modified dispersion in these tapers for ultra-low threshold supercontinuum generation.