The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number
$Re_\tau$
. This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of
$Re_\tau$
, owing to the natural constraint that the dissipation rate is bounded. Specifically,
$\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$
where
$\varPhi$
represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript
$\infty$
denotes the bounded asymptotic value of
$\varPhi$
, and the coefficient
$C_\varPhi$
depends on
$\varPhi$
but not on
$Re_\tau$
. Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form
$\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$
, where
$\varphi$
represents the streamwise or spanwise velocity fluctuation, and
$\alpha _q$
and
$\beta _q$
are independent of
$Re_\tau$
. Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear
$q$
-norm Gaussian’, is proposed to explain the observed linear dependence of
$\alpha _q$
on
$q$
.