I do a fair amount of amateur 'armchair thinking' about the n-body problem,
about why it hasn't been solved. (No solution from me yet-- ha!)

It seems that a new mathematical tool still needs to be found or invented, to
solve this and give an exact solution... a tool that's within our grasp, but
that just hasn't yet been formulated by a genius like Newton, Einstein or
Feynman.  (By solving it, I mean for the purely gravitational interactions of
all the planets, their moons, and the Sun-- without the 101 other subtle effects
involved.)

Calculus was invented to solve a 'similarly difficult' kind of problem (in
another mathematical realm), which had no real solution until then. And
Feynman's 'renormalization process' helped put quantum electrodynamics on a
surer footing. My gut feeling is that a new kind of tool like these is needed,
for exact solutions to mutually interacting objects, when there are lots of
objects involved.

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