What limits the doping concentration in a semiconductor? Si and Ge can be blended in any ratio, $\mathrm{Si}_x\mathrm{Ge}_{1-x},\ 0\le x\le 1$. So do 
InxGa1-x.
So what exactly causes doping impurities inside Si/Ge/etc. to saturate at $\sim 10^{-19}\ \mathrm{cm^{-3}}$?
 A: In the chemistry of liquids and solids, some combinations are "miscible", meaning they mix in any ratio (like water + alcohol or silicon + germanium). Other combinations are not (like water + oil or phosphorus + silicon), in which case there is a certain "solubility", and if you try to put in more than the "solubility limit" it prefers to segregate into separate phases (e.g., little nano-crystal inclusions can form).
Fun fact: You can actually put in more dopants than the solubility limit! ...But only if you use tricks like ion implantation + pulsed laser melting, where the solid freezes so fast that the molecules get stuck in place and cannot segregate even though segregating is the more stable configuration.
It's not terribly surprising that silicon & germanium are miscible, because they are both happiest in the exact same crystal structure (the diamond lattice), which in turn is because they have same number of valence electrons, i.e. the same column of the periodic table. For the same reason, they don't dope each other.
Likewise, it's not terribly surprising that silicon and phosphorus are not miscible because they do not prefer the same crystal structure. Phosphorus is happier when bonded to three neighbors, whereas it has four neighbors when it is shoehorned into silicon's diamond lattice. So if enough phosphorus is in the same place, they will rearrange into a different kind of lattice they like better.
A: These are actually two questions in one. On one hand, certain materials are miscible like for $$ Si_{1-x}Ge{x} $$ or likewise for $$ In_xGa_{1-x}As $$. Depending on the phase diagram, some materials can be mixed, while some are not soluble and would segregate, like Steve B mentioned before. Some materials can be mixed despite a miscibility gap, if the growth process does not take place at thermal equilibrium. The material is then frozen in its mixed form and depending on the energy barriers can not segregate at room temperature.
The second question is the doping limit, where doping means the introduction of "foreign" species. In fact what counts is not the dopant concentration but the electron or hole concentration. These saturate, depending on the material somewhere in the range between 3e18 and 2e19/cm³. If you put more dopants in, you will not necessarily get more free electrons. This can be due to self-compensation effects: Si in GaAs can go onto a Ga site and create an electron, which is usually favored, but at high concentrations, some Si atoms may also go onto As sites and therefore create holes. For P doping of Si, there are no different crystal sites, but saturation effects could occur due to P clustering, which also can lead to different behavior.
Doping behavior sometimes is not very intuitive to understand. Si in GaAs prefers to sit on the Ga site, although Si would be "larger" than Ga, but C prefers to go onto As sites, although C is "smaller" than Si and could therefore comfortably occupy a Ga site.
A: If you dope the semiconductor too much it becomes what is called degenerate. 
At normal levels of doping, the dopant atoms generate localized states in the material that can donate electrons or holes by either thermal excitation or optical excitation (e.g., a photon hitting a solar panel). The more you increase the doping the more likely it becomes that individual defect atoms will be closer together and merge into a midband (impurity state). At that point it stops behaving like a semiconductor because you've destroyed the necessary energy gap.
This basically means it becomes a bad metal, which by definition means that the conductivity of the material increases with temperature. 
