[staunton]

The crucial parts of that are "closed-form" and "standard". The analytic solution is "non-standard" because it involves the kind of power series that nobody knows or cares about (because they are only about 100 years old and have no real useful applications in engineering).

A similar claim is that roots of polynomials of degree 5 (and over) have no "general closed form solution" (with, as usual, the implicit qualification: "in terms of functions I'm currently comfortable with because I've seen them a lot"). That doesn't mean it's a difficult problem.

The two problems have in common that they are significantly harder than their smaller versions (two bodies, or degree 4). Historically, people spent a lot of time trying to find solutions for the larger problems in terms of the same functions that can be used to solve the smaller problems (conic sections, radicals). That turned out to not be possible. This is the historical origin of the meme "three body problem is unsolvable".

[Panzer04]

Ill probably go look this up, but do you mean functions of a higher type than normal powers like eg. Tetration, or something more complicated (am I even on the right track?)

[staunton]

I mean functions defined by power series (just like sin(x) is defined in analysis courses). For the three body problem, see http://oro.open.ac.uk/22440/2/Sundman_final.pdf (Warning, pdf!). This is what Wikipedia cites when talking about the solution to the three body problem. The document gives a lout of historical context.

For polynomial roots, see wikipedia.org/wiki/Elliptic_function.