The scaling of different features of streamwise normal stress profiles uu (+) (y(+)) in turbulent wall-bounded flows is the subject of a long-running debate. Particular points of contention are the scaling of the 'inner' and 'outer' peaks of < uu >(+) at y(+) (sic) 15 and y(+) = O(10(3)), respectively, their infinite Reynolds number limit, and the rate of logarithmic decay in the outer part of the flow. Inspired by the thought-provoking paper of Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, p. R3), two terms of an inner asymptotic expansion of uu (+) in the small parameter Re-tau(-1/4) are constructed from a set of direct numerical simulations (DNS) of channel flow. This inner expansion is for the first time matched through an overlap layer to an outer expansion, which not only fits the same set of channel DNS within 1.5% of the peak stress, but also provides a good match of laboratory data in pipes and the near-wall part of boundary layers, up to the highest Ret values of 105. The salient features of the new composite expansion are first, an inner < uu >(+) peak, which saturates at 11.3 and decreases as Re-tau(-1/4). This inner peak is followed by a short 'wall log law' with a slope that becomes positive for Ret beyond O(10(4)), leading up to an outer peak, followed by the logarithmic overlap layer with a negative slope going continuously to zero for Re-tau -> infinity.