A graph G is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are adjacent. For a non-Hamiltonian graph G, the Hamiltonian complete number H(G) is the minimum number of edges that must be added to G to make it Hamiltonian. We focus on both regular and irregular caterpillar graphs, deriving explicit formulas for H(G) in various cases. Specifically, we show that for a regular caterpillar graph Gn(k) where each vertex on the central path is adjacent to k leaves, H(Gn(k)) = n(k−1). We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for H(G) in these cases. Our results contribute to the understanding of Hamiltonian properties in tree-like structures and have potential applications in network design and optimization.