Why is the rendering equation, introduced by Kajiya in 1986, not solvable directly/analytically?
I'm sadly not able to add a comment to the answer above (not enough reputation), so I will do it like this.
I'd like to point out that what Dragonseel describes is simply an integral equation (specifically a Fredholm equation of the second kind). There are many such equations which do have an analytic solution; even some forms of the rendering equation have one (e.g. the solution of a white furnace can be given using a simple convergent geometric series, even though the rendering equation is infinitely recursive).
It is also not necessary to bias the estimated solution by bounding the number of recursions. Russian Roulette provides an useful tool for giving us an unbiased solution for an infinitely recursive rendering equation.
The main difficulty lies in the fact that the functions for reflectance (BRDF), emitted radiance and visibility are highly complex and often contain many discontinuities. In these cases there often is no analytic solution, or it is simply unfeasible to find such a solution. This is also true in the one dimensional case; most integrals lack analytic solutions.
Finally I'd like to note that even though most cases of the rendering equation do not have analytic solutions, there is a lot of research in forms of the rendering equation which do have an analytic solution. Using such solutions (as approximations) when possible can significantly reduce noise and can speed up rendering times.
The rendering equation is as follows:
Now, the integral is over the sphere around the point $x$. You integrate over some attenuated light, incoming from every direction.
But how much light comes in? This is the light $L(x',\omega_i)$ that some other point $x'$ reflects in the direction $\omega_i$ of point $x$.
Now you have to calculate how much light that new point $x'$ reflects, which requires solving the rendering equation for that point. And the solution for that point depends on a huge number of other points, including $x$.
In short, the rendering equation is infinitely recursive.
You cannot solve it exactly and analytically because it has infinite integrals over infinite integration domains.
But since light gets weaker each time it is reflected, at some point a human simply cannot notice the difference any more. And so you do not actually solve the rendering equation, but you limit the number of recursions (say reflections) to something that is 'close enough'.