  We prove existence and uniqueness of renormalized solutions to general
nonlinear parabolic equation in Musielak-Orlicz space avoiding growth
restrictions. Namely, we consider \[\partial_t u-\mathrm{div} A(x,\nabla u)=
f\in L^1(\Omega_T),\] on a Lipschitz bounded domain in $\mathbb{R}^n$. The
growth of the weakly monotone vector field $A$ is controlled by a generalized
nonhomogeneous and anisotropic $N$-function $M$. The approach does not require
any particular type of growth condition of $M$ or its conjugate $M^*$ (neither
$\Delta_2$, nor $\nabla_2$). The condition we impose on $M$ is continuity of
log-H\"older-type, which results in good approximation properties of the space.
However, the requirement of regularity can be skipped in the case of reflexive
spaces. The proof of the main results uses truncation ideas, the Young measures
methods and monotonicity arguments. Uniqueness results from the comparison
principle.
