  Scattering experiments with three free nucleons in the ingoing channel are
extremely challenging in terrestrial laboratories. Recently, the ALICE
Collaboration has successfully measured the scattering of three protons
indirectly, by using the femtoscopy method in high-energy proton-proton
collisions at the Large Hadron Collider. In order to establish a connection
with current and future measurements of femtoscopic three-particle correlation
functions, we analyse the scenarios involving $nnn$ and $ppp$ systems using the
hyperspherical adiabatic basis. The correlation function is a convolution of
the source function and the corresponding scattering wave function. The finite
size of the source allows for the use of the free scattering wave function in
most of the adiabatic channels except the lowest ones. The scattering wave
function has been computed using two different potential models: $(i)$ a
spin-dependent Gaussian potential with parameters fixed to reproduce the
scattering length and effective range and $(ii)$ the Argonne $v_{18}$
nucleon-nucleon interaction. Moreover, in the case of three protons, the
Coulomb interaction has been considered in its hypercentral form. The results
presented here have to be considered as a first step in the description of
three-particle correlation functions using the hyperspherical adiabatic basis,
opening the door to the investigation of other systems, such as the $pp\Lambda$
system. For completeness, the comparison with the measurement by the ALICE
Collaboration is shown assuming different values of the source radius.
