Note to Americans (Happy 4th of July by the way), that chilli is a hot spice you put on things. It's normally used to give food a kick. It's quite nice as the body reacts to this by releasing endorphins and you get a natural buzz.
Chilli Con Carne, is a (sometimes highly) spiced dish made of red peppers, ground meat, and often beans.
Now sure it's ok to called Chilli Con Carne, Chilli for short, when it's obvious what you are talking about. When it's not obvious then please specify the Con Carne bit. Really.
For example, a "cheese & chili" burger, one would expect to be some melted cheese and some endophine kicking chillis. One would not expect Chilli Con Carne. It makes no sense to have Chilli Con Carne on a burger. Because what you basically are making is a topping that is the actual burger.
Lets see.
Burger Patti = Ground beef packed together.
Chilli Con Carne = Ground beef not packed together.
So a CCC Burger is esentially a topping that is just a broken up burger. IT MAKES NO SENSE. It's like ordering a pizza with a topping of garlic bread.
Sigh. Apolgies if this isn't actually an American thing, but I've only had this problem in American restaurants.

1. Matthew
The topping is not really broken up burger though, as it was never a burger to begin with, so it couldn’t be broken up. [/pedantics]
Is food supposed to make sense?! I thought it was just there for the eating.
Err…pardon? I can only imagine the amount of phlegm and spittle you might find in the replacement burger.2. Rob
It’s always referred to as just Chili in North America (from what I’ve seen, at least). They’ll always use the full name for everything else (chili peppers, chili powder, etc).
I can see how you’d be surprised by what ended up on your plate, though. :)
3. razorhead
Chewbacca. Chewbacca is a Wookiee from the planet Kashyyyk, but Chewbacca lives on the planet Endor. Now, think about that. That does not make sense! Why would a Wookiee—an eight foot tall Wookiee—want to live on Endor with a bunch of two foot tall Ewoks? That does not make sense!
And now the Chilli Burger Defense. It does not make sense!