Super-exponential growth of the Poisson algebra in the planar three-body problem

We study the Lie algebra generated by the three pairwise interaction Hamiltonians of the planar Newtonian three-body problem under the Poisson bracket. Using exact symbolic computation with a polynomial representation of the inverse-distance potential, we determine the dimensions of the algebra through four bracket levels. The dimension sequence d(0)=3, d(1)=6, d(2)=17, d(3)=116 is proved exactly for levels 0-3, and a lower bound d(4) >= 4,501 is established numerically. The growth is super-exponential, implying infinite Gelfand-Kirillov dimension. The sequence is invariant under changes of mass ratios, including the exceptional Tsygvintsev cases where first-order Morales-Ramis obstructions vanish. Comparison with alternative potentials reveals a sharp structural dichotomy: the integrable harmonic potential (V ~ r^2) produces a finite-dimensional algebra (stabilising at dimension 15), while both the Newtonian (1/r) and Calogero-Moser (1/r^2) potentials yield the identical sequence 3, 6, 17, 116. The dimension sequence is thus a new algebraic invariant of the pairwise Hamiltonian decomposition, determined by the singularity class of the potential rather than by the coupling constants or integrability status.