  It is known, that the existence of dead ends (of arbitrary depth) in the
Cayley graph of a group depends on the chosen set of generators. Nevertheless
there exist many groups, which do not have dead ends of arbitrary depth with
respect to any set of generators. Partial results in this direction were
obtained by \v{S}uni\'c and by Warshall. We improve these results by showing
that abelian groups only have finitely many dead ends and that groups with more
than one end (in the sense of Hopf and Freudenthal) have only dead ends of
bounded depth. Only few examples of groups with unbounded dead end depth are
known. We show that the Houghton group \Hou with respect to a standard
generating set is a further example. In addition we introduce a stronger notion
of depth of a dead end, called strong depth. The Houghton group \Hou has
unbounded strong depth with respect to the same standard generating set.
