  The present paper represents an attempt for a very generic string inspired
theory of gravitation, based on a stringy action in the teleparallel gravity
which includes a specific functional which depends on the scalar field and its
kinetic energy, as well as the torsion and boundary terms, embedding also
possible effects from the teleparallel Gauss--Bonnet invariants. We focus our
study on FLRW cosmology. After we deduce the cosmological equations for the
associated generic theory of gravitation, we focus on string inspired couplings
which are studied by considering different analytical techniques. The first
analytical technique is based on the linear stability theory, by introducing
proper dimensionless variables which enables us to study the structure of the
phase space and the associated physical effects. In this case, we have obtained
different cosmological solutions which correspond to matter and dark energy
dominated solutions, achieving a possible transition between matter and dark
energy dominated epochs. For each type of cosmological solutions, we have
discussed the corresponding physical features, attaining viable constraints for
the coupling constants due to dynamical effects. The dynamical study of the
physical features included also a numerical analysis by fine--tuning the
initial conditions deep into the matter era, obtaining possible trajectories
for the effective equation of state for specific coupling functions.
