  We {\it analytically} compute, to the eight-and-a-half post-Newtonian order,
and to linear order in the mass ratio, the radial potential describing (within
the effective one-body formalism) the gravitational interaction of two bodies,
thereby extending previous analytic results. These results are obtained by
applying analytical gravitational self-force theory (for a particle in circular
orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant
redshift variable. We emphasize the increase in \lq\lq transcendentality" of
the numbers entering the post-Newtonian expansion coefficients as the order
increases, in particular we note the appearance of $\zeta(3)$ (as well as the
square of Euler's constant $\gamma$) starting at the seventh post-Newtonian
order. We study the convergence of the post-Newtonian expansion as the
expansion parameter $u=GM/(c^2r)$ leaves the weak-field domain $u\ll 1$ to
enter the strong field domain $u=O(1)$.
