This is really a great thread, my thanks to bunyupp for starting it. 
I can’t get enough of pop songs that make use of mandolins, dulcimers, accordions, baritone harmonicas, and brass instruments.
I like improvised instruments too.
The 1990s song Cannon Ball by the Breeders made use of a police whistle being blown into a bucket of water. They also did some pretty cool Ewok praying chant during the intro.
The German band Subway to Sally uses bagpipes, hurdy-gurdy, and other unusual instruments. This is a new song Schwarze Seide.
“Sweet Emotion” by Aerosmith has a bass marimba intro, played by Jay Messina.
It’s not quite in the same vein as the rest of the examples given, but Peter Gabriel’s work with the Fairlight CMI was profoundly non-traditional at the time.
The idea of sampling is old hat now, but once upon a time it was a profoundly new idea that you could turn anything in the world into an instrument by stretching/distorting its sound. My favorite example is Gabriel’s San Jacinto – that opening chime/bell section is actually banging on junkyard pipes.
Paul Simon used pan pipes (pan flute) on at least two recordings: Lincoln Duncan and El Condor Pasa.
Fleetwood Mac used a high school marching band on Tusk.
And of course, The Moody Blues used a symphony orchestra.
The short-lived 80’s band Pianosaurus played toy instruments.
All female metal band from Mexico “Mystica Girls” features the lead singer also playing violin.