Silicon purity and electronic feature density
That's a surprisingly involved question.

So, first of all, the critical thing for commercially available microprocessors is reliable, well-defined transistors and diodes, based on pn-junctions.

Reliable implies that under no circumstances should a semiconductor junction act like a metal. However, all semiconductors do that as soon as you dope them too heavily. 

This implies that for commercial semiconductor devices, you need so-called lightly doped semiconductors. Obviously, you want the doping atoms to be the dominant non-Silicon atoms - so every other kind of atom, lest it be really insignificant for the bands involved, must be at least two or three orders of magnitude rarer.

So, what is a lightly-doped semiconductor? Definitions differ, but the takeaway is that the semiconductor must not be degenerate, that is

$$\begin{align}
E_c - \mu &amp;\gg k_BT \\
\mu - E_v &amp;\gg k_BT \\
\end{align}$$
must hold. 

Here, \$E_c\$ and \$E_v\$ are the energies in the conducting and valence bands, each, which are fixed for a given semiconductor. \$\mu\$ however, is the so-called Fermi-level, which is the level of "occupancy probability equilibrium". In other words, at room temperature, an electron is as likely as not (50%) to be at that level of energy. What the inequalities above demand is that the distance from that level to the band edges is big enough to avoid charges to suddenly flow spontaneously.

\$\mu\$ depends on the doping, since it is a function of the number of charge carriers. Now, I'm not an expert on semiconductor production itself, but I'd assume that with typical doting in the region of \$10^{-15}\text{cm}^{-3}\$, and the molar mass of silicon, you can get far enough to get an idea of what's technically necessary for pure silicon-based bijunction transistors. Then, add some security margin – structures are very small, so a cluster of lets say 3 unwanted atoms might already be pretty much, so the likelihood of that must be suppressed sufficiently below your acceptable yield. It boils down to a size/cost/reliability/yield trade-off and no general answer can be given.

Now, these calculations are based on the assumption that we're working on a silicon wafer with the "natural" silicon lattice – and in reality, modern ICs are based on SOI – silicon stretched on an isolator (SiO2, typically). It's very hard to actually talk about purity in this context.