  We explore \emph{semibounded} expansions of arbitrary ordered groups; namely,
expansions that do not define a field on the whole universe. We introduce the
notion of a \emph{semibounded} expansion of an arbitrary ordered group,
extending the usual notion from the o-minimal setting. For $\mathcal{R}=(
\mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq
\mathbb{R}$ a set satisfying certain tameness conditions, we discuss under
which conditions $(\mathcal R,P)$ defines total linear functions that are not
definable in \mathcal{R}. Examples of such structures that does define new
total linear functions include the cases when $\mathcal{R}$ is a reduct of
$(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda
x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an
iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.
