$|x| \ll 1$ is understood as "$x$ sufficiently small (in norm)", i.e. $|x|<C$ for some unspecified constant $C$. Similarly, $|x| \gg 1$ is understood as "$x$ sufficiently big".
Has anyone used the notation $a\ll -1$ for $a\in\mathbb R$ sufficiently large and negative, and is there a reason it could be misleading (i.e. ignoring aesthetics)?