  We consider exclusive multi-peripheral production of four charged pions in
proton-proton collisions at high energies with simultaneous exchange of three
pomerons/reggeons. The amplitude(s) for the genuine $2 \to 6$ process are
written in the Regge approach. The calculation is performed with the help of
the GenEx Monte Carlo code. Some corrections at low invariant masses in the
two-body subsystems are necessary for application of the Regge formalism. We
estimate the corresponding cross section and present differential distributions
in rapidity, transverse momenta and two- and four-pion invariant masses. The
cross section and the distributions depend on the value of the cut-off
parameter of a form factor correcting amplitudes for off-shellness of
$t$-channel pions. Rather large cross section is found for the whole phase
space ($\sigma \sim$ 1-5 $\mu$b, including absorption corrections). Relatively
large four-pion invariant masses are populated in the considered diffractive
mechanism compared to other mechanisms discussed so far in the context of
four-pion production. We investigate whether the triple Regge exchange
processes could be identified with the existing LHC detectors. We consider the
case of ATLAS and ALICE cuts. The ATLAS (or CMS) has better chances to identify
the process in the region of large invariant masses $M_{4 \pi} > 10$ GeV. In
the case of the ALICE experiment the considered mechanism competes with other
mechanisms (production of $\sigma \sigma$, $\rho \rho$ pairs or single
resonances) and cannot be unambiguously identified.
