In the quest to understand high-temperature superconductivity in copper oxides, a vigorous debate has been focused on the pseudogap-a partial gap that opens over portions of the Fermi surface in the 'normal' state above the bulk critical temperature (Tc). 1 The pseudogap has been attributed to precursor superconductivity, to the existence of preformed pairs, or to competing orders such as charge-density waves. 1-4 A direct determination of the charge of carriers as a function of temperature and bias could help resolve among these alternatives. Here, we report measurements of the shot noise of tunneling current in high-quality La2-xSrxCuO4/La2CuO4/La2-xSrxCuO4 (LSCO/LCO/LSCO) heterostructures fabricated using atomic-layer-by-layer molecular beam epitaxy, for several doping levels. The data delineate three distinct regions in the bias voltage-temperature (V-T) space. Well outside the superconducting gap region, the shot noise agrees quantitatively with independent tunneling of charge-e carriers. Deep within the gap, shot noise is greatly enhanced, reminiscent of multiple Andreev reflections. 5-7 Starting above Tc and extending to biases much larger than the gap, there is a broad region in which the noise substantially exceeds the expectations of single-charge tunneling, indicating pairing of carriers. Pairs are detectable deep into the pseudogap region of temperature and bias.