Is a three body gravitating system doomed to collapse?
Since you are talking about real bodies with some extent, yes. The case for point-particles is somewhat different.
Any dissipative system will (by definition) gradually lose energy, and that means that the positions and velocities of the constituent bodies will be confined to (hyper)surfaces in phase space with less and less energy, $E=V(\mathbf{x})+K(\mathbf{v})$. Potential energy $V(\mathbf{x})$ decreases with distance between the bodies and there is a minimum possible separation giving a minimum potential energy, and kinetic energy is always positive $K(\mathbf{v})&gt;0$. So as $E$ declines the distances and velocities have to decline accordingly. Typically things getting close move faster, with kinetic energy increasing as potential energy decreases, but since there is a limit to how much potential energy can decrease before things collide there will eventually be a collision (or, in systems allowing it - not the case here - losing all kinetic energy and ending up in perpetual rest).
(Point particles can cheat in Newtonian gravity by performing weird and physically unrealistic collision or oscillating orbits since they can reduce their potential energy arbitrarily far, see the Painleve conjecture)
In practice dissipation effects are tiny in planetary systems, and for most planets their orbits will instead be disrupted by rare stellar encounters, chaotic resonances with other planets, or mass-loss from the star when it leaves the main sequence. Planets near stars may be dragged in because of tidal effects, but this matters most if one is a few stellar radii away. In the truly long run gravitational radiation will force any planet to spiral in, but the timescales are staggeringly long.
The number of bodies does not change things, but for many-body systems ($N\geq 3$) some bodies can gain enough energy to permanently escape while the rest merge.