Suppose you have a matrix of 2×2 copper electrodes per millimeter. So if you have a 100×100 millimeter area, you might have 200×200 electrodes, a total of 40,000. To each you have attached a transistor. On top of this matrix, you have a bath of electrolyte containing, for example, copper salts. If you lay a sheet of metal atop this bath of electrolyte at a depth of less than half a millimeter or so, you can selectively copper-plate certain parts of the sheet, printing a copper pattern on the sheet, by passing current through some of the electrodes and not others.
You can renew the electrodes to some extent by taking the workpiece out, replacing it with a sacrificial copper sheet, and reversing the current.
With this approach, you could conceivably electrodeposit one layer after another of copper until you have generated whatever shape you desire, gradually lifting the workpiece out of the bath. However, copper plating is a slow process, with deposition speeds of tens of microns per hour. The very short distance traveled by the current in this approach might allow the process to run a little faster.
This suggests that this approach might be more practical at a smaller scale: if your deposition rate is 10 microns per hour, that’s about 2.8 nm per second. So you could reasonably space your electrodes 500 microns apart and electrodeposit your copper in 500-nm layers, one every three minutes. If you are doing this over a 10mm×10mm square chip, each layer contains 400 million voxels, or about 2 million per second, which is an eminently feasible data rate. Filling a cube in this way would require 1000 hours (plus the time to refresh the electrodes), so you probably want to make thin things instead.
The ability to deposit multiple different materials, such as copper and zinc, or copper and nickel, or gold and nickel, or nickel and tin, or copper and chromium, or lead and chromium, seems like it would dramatically extend the reach of this technique, allowing the construction of metamaterials with vanishing thermal coefficients of expansion, high-current-density batteries, or bimetallic structures that deform in predetermined ways according to temperature, for example.
Material cost is likely to not be an issue, except for extremely exotic materials and maybe gold. A milliliter of gold weighs about 0.6 troy ounces, or about US$1000; above I estimated that the device would need 1000 hours to deposit it, using about US$1 of gold per hour, but probably costing significantly more than US$1 per hour in labor to operate. Other normal metals cost orders of magnitudes less. Extremely exotic materials might include things like pure lanthanides (as opposed to Mischmetall) and isotopically pure metals.
Using a molten-salt electrolyte like the Hall-Héroult cell, rather than an ionic-solution electrolyte, might allow electrodeposition that is more favorable in one or another way: for example, you could electrodeposit aluminum and other metals that react too easily with water.
Of course, the ability to selectively electrodeposit controllably doped semiconductors in this way would be immensely valuable, but I don’t know of any semiconductors that would be viable candidates by themselves. It might be possible, though, to selectively electrodeposit a doped metal, then oxidize the metal into a semiconductor by exposing it to an oxidizer, maybe at high temperature. For example, you could oxidize zinc to zinc oxide or zinc sulfide, lead to lead sulfide (galena), copper to copper oxide, or titanium to titanium dioxide.