  In this paper, we study the one-dimensional cubic nonlinear Schr\"odinger
equation (NLS) on the circle. In particular, we develop a normal form approach
to study NLS in almost critical Fourier-Lebesgue spaces. By applying an
infinite iteration of normal form reductions introduced by the first author
with Z. Guo and S. Kwon (2013), we derive a normal form equation which is
equivalent to the renormalized cubic NLS for regular solutions. For rough
functions, the normal form equation behaves better than the renormalized cubic
NLS, thus providing a further renormalization of the cubic NLS. We then prove
that this normal form equation is unconditionally globally well-posed in the
Fourier-Lebesgue spaces $\mathcal{F} L^p(\mathbb{T})$, $1 \leq p < \infty$. By
inverting the transformation, we conclude global well-posedness of the
renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable
sense. This approach also allows us to prove unconditional uniqueness of the
(renormalized) cubic NLS in $\mathcal{F} L^p(\mathbb{T})$ for $1\leq p \leq
\frac32$.
