  In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and
their small perturbations: $dF+\epsilon\omega=0$. The first nonzero Melnikov
function $M_{\mu}=M_{\mu}(F,\gamma,\omega)$ of the
  Poincar\'e map along a loop $\gamma$ of $dF=0$ is given by an iterated
integral. In a previous work (see arXiv 1703.03837), we bounded the length of
the iterated integral $M_\mu$ by a geometric number $k=k(F,\gamma)$ which we
call orbit depth. We conjectured that the bound is optimal. Here, we give a
simple example of a Hamiltonian system $F$ and its orbit $\gamma$ having
infinite orbit depth. If our conjecture is true, for this example there should
exist deformations $dF+\epsilon\omega$ with arbitrary high length first nonzero
Melnikov function $M_\mu$ along $\gamma$. We construct deformations
$dF+\epsilon\omega=0$ whose first nonzero Melnikov function $M_\mu$ is of
length three and explain the difficulties in constructing deformations having
high length first nonzero Melnikov functions $M_\mu$.
