We report on the experimental demonstration of refractive micro-optical elements with arbitrary exponential surface profiles. Refractive optical elements such as lenses and axicons have parabolic (power-exponent of two) or conical (power-exponent of one) surface profiles, respectively. Here, we analyze micro-optical elements with non-parabolic surface profiles characterized by both integer and fractional power-exponents between 0.5 and 3.25. The experimentally measured beam shaping properties of the components were compared with numerical simulations and shown to be in excellent agreement. The generalized family of optical elements with non-parabolic power-exponent elements allows variation of the optical performance within the same device form-factor and enhancement of various optical properties, such as the extension of the depth-of-focus or resolution improvement.