  We show that real semi-simple Lie groups of higher rank contain (infinitely
generated) discrete subgroups with full limit sets in the corresponding
Furstenberg boundaries. Additionally, we provide criteria under which discrete
subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set
in the Furstenberg boundary of $G$.
  In the appendix, we show the the existence of Zariski-dense discrete
subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such
that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies
on the boundary of the limit cone of $\Gamma$.
