  We discuss the production of $t\bar t$ quark-antiquark pairs in proton-proton
collisions via the $\gamma\gamma$ fusion mechanism. We include topologies in
which both protons stay intact or one or even both of them undergo
dissociation. The calculations are performed within the $k_{\rm
T}$-factorisation approach, including transverse momenta of intermediate
photons. Photon fluxes associated to inelastic (dissociative) processes are
calculated based on modern parameterisations of proton structure functions. We
find an integrated cross section of about 2.36 fb at $\sqrt{s}=$ 13 TeV for all
contributions. The cross section for the fully elastic process is the smallest.
Inelastic contributions are significantly reduced when a veto on outgoing jets
is imposed. We present several differential distributions in rapidity and
transverse momenta of single $t$ or $\bar t$ quarks/antiquarks as well as
distributions in invariant mass of both the $t\bar t$ and masses of dissociated
systems. A few two-dimensional distributions are presented in addition.
