In this paper we study polynomial Hamiltonian systems dF=0 in the plane and their small perturbations: dF+ω=0. The first nonzero Melnikov function Mμ=Mμ(F, γ, ω) of the Poincaré map along a loop γof dF=0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral Mμby a geometric number k=k(F, γ) which we call orbit depth. We conjectured that the bound is optimal.Here, we give a simple example of a Hamiltonian system Fand its orbit γhaving infinite orbit depth. If our conjecture is true, for this example there should exist deformations dF+ω with arbitrary high length first nonzero Melnikov function Mμalong γ. We construct deformations dF+ω=0 whose first nonzero Melnikov function Mμis of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.