Chicago suburbs, and as has been mentioned: RC Cola. It’s a dollar for 2 liters at my liquor store, and it’s a fine cola.
Not sure there is one here aside from store brands. I remember this one growing up in NZ (apparently an Australian brand).
My local soft drink company back then was West’s Cordials. These original flavours (chocolade; pineapple-and-pear) are worth your while if you’re ever in my home town.
Here in Tel Aviv, we don’t even have a second cola. It’s just Coke all the way.
Here in the PNW, Jones is probably the closest thing to a third brand that we have.
Beijing. The big two here are, of course, Coca-Cola and Pepsic Cola. Another popular option is Arctic Ocean soda (aka Polar Bear), a Chinese brand which I quite like. Fairly popular also are Watson’s various flavors. Watson’s is a Hong Kong health care and beauty chain. The ones I like the most from that chain are Vanilla, Ginger Ale, and Sarsae (a sarsparilla soda). Their Salt Soda is okay and the Lemongrass Soda is just vile.
I can’t recall the last time I saw RC in a store. Apart from the big 2, all I see are store brand colas.
Cleveland area, and I suppose it’s RC, but you’re much more likely to see the various store brands, and the third-biggest seller might well be one of those store brands.
In Cuba, there was a quite tasty and invigorating cola called Tukola, which was basically the only brand since they weren’t allowed to carry American brands like Coke or Pepsi. One of my students figured I liked it so much because he said it probably had cocaine in it.
I live in the Atlanta area, where there is a question as to what is the ‘second cola’ 
Here in NC we have Cheerwine which is a cherry flavored cola and is popular.
Rural western Pennsylvania, after Pepsi and Coke, Generic Cola comes in third (assuming Sprite and Mountain Dew do not count).
I was going to say probably RC here (Northern California) if only because it’s the only other one sold at Target along side the big two (Target doesn’t have a store brand cola). But it wouldn’t surprise me if it was one of the big store brands. Without any actual market share data it’s really just a guess.
Oddly, when I want to an airshow here years ago, all the concession vendors were selling RC Cola. I always wondered how they managed to get that contract over Coke and Pepsi.
You’d think Shasta Cola would be bigger here, after all it is named after a huge dormant volcano in Northern California, and the company’s headquarters is in the region. But I only ever seem to see it in dollar stores.
I grew up in NC, and IIRC Cheerwine isn’t a “cola”. It’s a black cherry flavored soda.
In the days of my youth in SoCal, it was definitely RC cola (my soda of choice, because it came in a 16-ounce bottle, unlike the 12-ounce chintzy-ass microportions from Coke and Pepsi). I don’t tend to haunt the soda aisle at the supermarket these days,* but I can’t recall seeing RC in several years.
*Nowadays, the #3 cola (AFAIAC) is Shasta, but that feels like a store brand (even though I know it’s not).
Vivi Kola, developed in Switzerland in 1938 , along with Fritz-Kola developed in Hamburg in 2002 are the most common colas in Switzerland, besides Pepsi and Coke.
Based on the amount of shelf space they occupy at the local grocery, I’d say around here it goes: Coke, Pepsi, Shasta. I’m including Mexican Coke in with all the others by Coca-Cola. RC is around, but not all that popular.
Pretty sure Soldier Field (Chicago Bears) only has RC Cola as well.
RC was pretty common in NYC in the nineties. It was the cola served at Yankee Stadium for at least a year or two. I don’t drink soda much anymore, so I’m not sure if it’s available in the DC area.
I didn’t know that about Yankee Stadium.
But when I was a kid in NYC, in the late sixties/early seventies, you could buy RC pretty much anywhere.
It’s gone – I haven’t seen it here in years. Decades, even.
I think I remember it still being sold when I took the kids to The House That Ruth Built the last year it existed, but I wouldn’t swear to it. I didn’t see it when I went to the new stadium a few years ago.
Texas here, and as others have said there really is no “third cola”. Maybe H‑E‑B has a house brand?