  In this work we propose the $f(Q,T_{\mu\nu}T^{\mu\nu})$ gravity as a further
extension of the $f(Q)$ and $f(Q,T)$ gravity theories. The action involves an
arbitrary function of the non-metricity $Q$ and $T_{\mu\nu}T^{\mu\nu}$ in the
gravity Lagrangian. The field equations for the theory are derived in the
metric-affine formalism by varying the action with respect to the metric. The
theory involves a non-minimal coupling between the geometric and the matter
sectors, and hence the covariant divergence of the energy-momentum tensor is
non-zero, thus implying the non-conservation of the same. The vacuum solutions
of the theory are investigated and it is found that the theory perfectly admits
a de-Sitter-like evolution of the universe. The cosmological equations are
derived and it is found that there are two correction terms arising as
modification of the gravity. Two specific toy models of the form $Q+\eta
\left(\textbf{T}^2\right)^{n}$ and $f_{0}Q^{m} \left(\textbf{T}^2\right)^{n}$
are explored to gain further insights into the dynamics of the theory. It is
seen that the field equations of both the models have terms similar to those
arising from the quantum gravity effects and are thus responsible for the
avoidance of singularity. One striking feature of the model is that the
non-linear correction terms dominate in the early universe and gradually fade
away at later times giving the standard FLRW universe. Solutions for the FLRW
equations are found wherever possible and the evolution of the scale factor and
the matter-energy density are plotted. Finally, the energy conditions are
explored in the background of the theory. Using these conditions and some
observational data the theory is considerably constrained.
