  The class, denoted by $\mathscr{S}$, of totally disconnected locally compact
groups which are non-discrete, compactly generated, and topologically simple
contains many compelling examples. In recent years, a general theory for these
groups, which studies the interaction between the compact open subgroups and
the global structure, has emerged. In this article, we study the non-discrete
totally disconnected locally compact groups $H$ that admit a continuous
embedding with dense image into some $G\in \mathscr{S}$; that is, we consider
the dense locally compact subgroups of groups $G\in \mathscr{S}$. We identify a
class $\mathscr{R}$ of almost simple groups which properly contains
$\mathscr{S}$ and is moreover stable under passing to a non-discrete dense
locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same
properties previously obtained for $\mathscr{S}$ and establish various original
results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$,
notably concerning the structure of the local Sylow subgroups and the full
automorphism group.
