Calling all Alda groupies! Apparently the prop moonshine still that Hawkeye and Trapper lovingly tended to over M*A*S*H's eleven seasons still exists somewhere in the Smithsonian National Museum of American History.
This meticulously detailed M*A*S*H fansite (of course these things exist) picked out every episode in which the still played a large part. A commenter noted that, even if the model of the still did work, it would be pretty inefficient; the condenser coils are fairly short, and they aren't submerged in a coolant, so much of the alcohol wouldn't make it back down to become Hawkeye's "breakfast of champions."
The still entered the Smithsonian collection after a two-year show at the museum following the M*A*S*H finale, and is, in all likelihood, collecting dust somewhere next to the Ark of the Covenant.
[M*A*S*H 4077 via BoingBoing]