The dead-end depth of an element
g
g
of a group
G
G
, with respect to a generating set
A
\mathcal {A}
, is the distance from
g
g
to the complement of the radius
d
A
(
1
,
g
)
d_{\mathcal {A}}(1,g)
closed ball, in the word metric
d
A
d_{\mathcal {A}}
defined with respect to
A
\mathcal {A}
. We exhibit a finitely presented group
G
G
with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.