Has anyone said “bay leaf” yet?
I don’t see it, so:
bay leaf.
I hate getting that frikin’ leaf.
Bird shot.
The really big roach that ran on top of it just as you were hoisting the platter in the kitchen.
Yeah, add cloves to the list. Ham tastes better with them, but not with them if you know what I mean.
Parsley. It’s a garnish, not an essential part of cleaning one’s plate, and my mom never knew that.
Mushrooms. Just my opinion, but while mushrooms do add to the flavor of chicken, I don’t like the slimy, chlorine-y tasting objects in my mouth.
Cyanogenic glucosides (in cassava).
Just the OP. 
Onion skins. I learned that a little too late… during my first college sit-down dinner. It was comical to watch my roommate quizically pull onion skin off her tongue.
Quit picking on the pitiful bayleaf! In our home, it’s tradition to leave it in the dish and whoever finds it is automatically excused from dishes duty that night. See? Nice tradition started from one of my earlier screwups. You folks have got to learn to WORK these mistakes to your advantage! 
Star anise
Large pieces of lemongrass
Whole lime leaves
In fact if you want a wealth of inedible items that enhance a soup get some Tom Kha Gai thai soup, me and my work friends make a little game to see wyho gets the most inedible pieces in their soup.
The cook’s thumb.
Lemme guess, you also put pennies in cupcakes as a good luck game? 
Heh, that’s the spirit!
Not in cupcakes, but definitely a few coins get popped into birthday cakes inmy family.
Philistine. Plebe! You don’t know the meaning of ‘haute cuisine’!
That said, cloves. I loathe cloves.
All these posts and no one has mentioned eggshells? Unless they’re hard-boiled, of course.
And don’t repeat my great-aunt’s mistake of canning peaches with the worms still in them. She wasn’t alone though; I once opened a store-bought can of peaches and found a little white larva, very dead, floating on top.
Sausage “casings” that are supposed to be removed.
Which has caused me a few quandries over the years - sometimes it’s tricky to discern just by eyeballing it whether the casing on a particular item is supposed to be edible.
Unless you’re having gagh!
There’s also a kind of fish, I forget the name, which in Japan is served live. Carefully sliced (diagonally up the flanks), but live. And wiggling on the plate.
Oysters are live when you eat them raw straight from the shell, aren’t they?
the little green threads on the sides of snow pea pods (as well as the stems)…
Ginger slices. shudder
Regarding the live fish, that sounds a bit like an UL. Cite?