Growing Spines Ad Infinitum

We show that every non-trivial ordered abelian group $G$ is augmentable by infinite elements, i.e., we have $G\preccurlyeq H\oplus G$ for some non-trivial ordered abelian group $H$. As an application, we show that when $k$ is a field of characteristic 0, then $k$ is not $t$-henselian if and only if all henselian valuations with residue field $k$ are ($\mathrm{\varnothing }$-)definable.