Unable to properly wire timer to old light switch with one “white?” Wire on top and one looped black wire
The double wire is simply power being carried onward to other switches.  

Note the solitary black wire.   Follow it to see which cable that wire departs the box in.  Now look at that cable's white wire and follow it back to a bundle of all-whites... Add your smarts witch neutral to that bundle.   Do not remove any wires from the bundle, they are not spares.  

Do not add it to any random bundle of whites you see. That will "seem to work" until you add a GFCI or AFCI, then will trip it.   Add it to the right bundle.