We all know that Captain Kirk is the Top Dog, the Head Cheese, Número Uno on the Enterprise. What crew member would be the bottom of the barrel in his/her rank and duty station? Use your imagination and sense of humor.
Crewman Apprentice 3rd Class Schmidlap, who duty was to window clean all of the interstellar bugs off of the portholes and view screens....from the OUTSIDE.
I know it's stating the obvious, but all the "Crewman Smiths" who went on away missions just to get killed.
On all Starfleet vessels, the lowest ranking officer traditionally holds the ceremonial title of "Ensign Kim."
Apprentice Spaceman Oliver Pippin (non-commissioned). He answers to the flight control officer, and his primary job is to keep the shuttles ready for launch. Not the engines or systems - *that* responsibility falls to engineers. I mean wash, wax, detailing, vacuuming, removing trash that may have been left aboard during last use, etc.
Pretty simply, whoever is the lowest rank (Crewman 3rd class), with the lowest seniority. By position, it could be the junior crewman on the janitorial team's gamma watch.
I guess I always thought that it was a crewman dressed in work clothes, walking through ship corridors as in the overly dramatic red alert in Where No Man Has Gone Before such as at 2:30 and 2:55 of the following. https://www.youtube.com/watch?v=Z68ixSPjoTo
Either the guy who cleaned out the antimatter chamber or the one who maintained the saucer's landing gear.
What about the lowly engineering technician who had to change all of the burned out light bulbs on the endless number of control panels on the bridge?