  A classical tool in the study of real closed fields are the fields $K((G))$
of generalized power series (i.e., formal sums with well-ordered support) with
coefficients in a field $K$ of characteristic 0 and exponents in an ordered
abelian group $G$. In this paper we enlarge the family of ordinals $\alpha$ of
non-additively principal Cantor degree for which $K((\mathbb{R}^{\le 0}))$
admits irreducibles of order type $\alpha$ far beyond $\alpha=\omega^2 $ and
$\alpha = \omega^3$ known prior to this work.
