So this is a fun question:
What is the instrument that you enjoy the most when present at a session? Doesn’t even have to be anything unique. Maybe there’s a particular sound not heard often that really punches in when played well, or maybe there’s a more ubiquitous instrument that you would sorely miss were a representative not there.
Any takers?
Flute (duh!) 
PIPES! What else is there?
Any session I’m in will have pipes by default. Therefore, I would choose fiddle with accordion as a close second.
Bedside pipes, tenor banjo,flute,fiddle,squeezeeee boxes, and whistles, and then some rhythm.
Few sessions near me have flute players, there’s always fiddlers and accordion players and loads of guitars and other stringed monsters so when I bring my flute it feels appreciated, just as well cos I dont have an accordion anymore.
Aside from my own instrument (the flute), I really enjoy when button accordion players show up to the session, and I feel the world needs more of them.
Flute and whistle… and I also really enjoy playing tenor banjo.
Whistle.
I was going to say hurdy-gurdy, because I love French dance music and really enjoy playing it with people who know how to punch the rhythms out on a gurdy.
But actually I think I’m actually fairly instrument-agnostic; so long as everyone’s acceptably in tune and acceptably competent, I think the best sessions are those where there’s a selection of instruments, with no more than two or three of any one particular instrument.
I was also going to say that I couldn’t think of an instrument that I wouldn’t welcome in a session, but the guy who turned up to our predominantly G/D session with a didgeridoo pitched somewhere around the Eb region made a, er, unique contribution …
Whistle.
Seems like Irish Traditional Music is creeping into World Music where anything goes with any instrument/implement.
Spoons!
I’m not sure that I’ve ever heard a didge make a welcome contribution to anything, frankly. Hypothetically, I suppose, it’d be excellent punctuation for a f*rt joke.
Guinness…
Best wishes.
Steve
I have a Chieftains record with a didgeridoo coming in on one of the later repetitions of The Strayaway Child and I think it sounds good; makes it kind of spooky. Don’t remember who plays it though. I’ll grant there’s not many other tunes I’d want to hear it on though.
As far as the original question goes, I love to see/hear a button accordion in a session.
When the Washboard and Gut Bucket Twins show up, I know things are going to get janky.
Singers.
The first instrument. The only one common to every musical tradition.
So under-represented at sessions…
Mandolin
Fiddle, which is possibly odd since I don’t play fiddle but can play whistle, flute, banjo, mandolin, and bouzouki. I often think maybe it’s BECAUSE I don’t play it that it’s my favorite.
I’m with you Thomaston. I only want to play the flute but good fiddle playing bends my heart.