  Recently, we presented a two-dimensional (2D) model of a weak topological
insulator formed by stacking an $N$ number of Su-Schrieffer-Heeger (SSH) chains
\cite{Agrawal_2022-02}. We now study the influence of periodic driving on the
topological properties of this system, which has all the fundamental
symmetries, by shining it with circularly polarized light (CPL). The CPL is
chosen because it breaks the time-reversal symmetry, which induces more exotic
topological properties in the system. We investigate two different formations
of the $N$ stacked SSH chains: all the SSH chains are topologically trivial in
one formation and nontrivial in the other one. In contrast to the undriven or
static case, both formations exhibit distinct topological behaviors. Here, we
particularly derive the Floquet or the effective Hamiltonian using the replica
method, which facilitates the study of high- and low-frequency regimes. We have
discovered that this model exhibits laser-induced Floquet topological phases
with higher Chern numbers. This system has nonlinear dispersion along both
directions with additional $k_x-k_y$ coupling terms, which made the dispersion
of this system {\it unconventional}. We closely study the role of this
unconventional dispersion in the system at the low-energy limit and its
response to periodic driving. The low-energy Hamiltonian also reveals a
hierarchy in the gaps of the neighboring Floquet bands. Interestingly, though
this model has nonlinear quasi-energy dispersion, it still shows some
signatures of hierarchy, which was observed in the system with linear
dispersion like graphene. Furthermore, we study the effect of linearly
polarized light (LPL) on the topological properties of the system. In response
to the LPL driving, the band-touching point either opens up or splits into two
band-touching points.
