  Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$
and any integers $K$ and $N$, we show that there exists a metric on $\mathcal
M$ for which the first $K$ nonconstant eigenfunctions of the
Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal
components. This provides a negative answer to the question of whether the
number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a
Courant-type bound, which has been featured in recent surveys by Girouard and
Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9,
Open question 10.14].
