  We provide a family of isolated tangent to the identity germs
$f:(\mathbb{C}^3,0) \to (\mathbb{C}^3,0)$ which possess only degenerate
characteristic directions, and for which the lift of $f$ to any modification
(with suitable properties) has only degenerate characteristic directions. This
is in sharp contrast with the situation in dimension $2$, where any isolated
tangent to the identity germ $f$ admits a modification where the lift of $f$
has a non-degenerate characteristic direction. We compare this situation with
the resolution of singularities of the infinitesimal generator of $f$, showing
that this phenomenon is not related to the non-existence of complex
separatrices for vector fields of Gomez-Mont and Luengo. Finally, we describe
the set of formal $f$-invariant curves, and the associated parabolic manifolds,
using the techniques recently developed by L\'opez-Hernanz, Raissy, Rib\'on,
Sanz S\'anchez, Vivas.
