Do you think it's possible? = Not really.
Has such a thing ever been proposed or prototyped? = actually yes, nearly 30 yrs ago. However, practicality is close to nil. You need insulators and metallization, which means you need also plasma/reactive ion etch, sputtering, passivation, annealing, etc. The closest thing to this are the active matrix transistor technology for displays. Hypothetically, one could do something like that with what amounts to an inkjet printer. But to what end? The transistors made with this type of process are barely good enough to crank a display, and nowhere near good enough to build anything with more restrictive requirements.
Power transistors, of any ilk, require precision depositions and anneals, as well as extremely good metallization. Processors require at least 2 layers of metallization, and more likely, 3 layers. An processor of any tolerable performance would require millions of transistors; doing them one by one would be equivalent to watching grass grow. Actually, grass would be faster. 5 million transistors at 10 seconds a pop = 1.58 yrs.
What would the major stumbling blocks be? where to begin? At the minimum, an office desktop system is physically impossible, unless the desktop was at least 100 sq ft in area, and about 8 ft tall:
High vacuum is required for ion implant, as well as plasma etch, as well as metallization. The pumps required to do high vacuum alone would occupy about half of a standard office desktop.
You could potentially get away with using only a laser anneal, but the laser for something like this would occupy at least one desktop's area. Rapid thermal anneal is more likely to be required, again, another desktop's worth of area. Plasma and reactive ion, as well as sputtering require rather massive RF power supplies; another desktop chewed up. A normal ion implanter is rather massive to begin with. The extraction supply alone would take up have a desktop. Direct write on wafer was proposed and demonstrated with e-beam lithography about 40 yrs ago. That alone was another desktop's area gone. I think I've already gone past my initial 100 sq. ft. area estimate.
The fact that you've apparently done zero research prior to posing this question is problematic.
TTFN
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