Laos is very much in my awareness, mostly because a friend spent a lot of time there last year.
Armenia doesn’t get much press. A few years ago I went on a date with an Armenian princess. Honestly, if she was the best they could come up with for a princess, no wonder they’re obscure.
Hoards.
And clusterfucks.
Really? They just had a coup and the north of the country is under rebel control.
My vote is for Mozambique. I was there a few months ago, but not too many people I’ve met can tell you anything about it.
Not a Sporcle fan, I take it? 
The Central African Republic is pretty darn obscure. Even the name sounds like a joke: a generic placeholder, like “Middle Eastern Warzone” or “South American Dictatorship”. But it’s bigger than Spain, and Germany, and Japan.
Actually, Indonesia has been in the news a little bit over the last few years because President Obama spent some of his childhood living there, and he has a half-sister who was born there. Obama made an official visit there during his first term in which he visited his childhood home, if I recall correctly.
Tanzania was in the news a few years ago when the U.S. embassy there was bombed. On the same day as the embassy in Nairobi, Kenya was bombed.
From that list, the biggest country that I know nothing about is Uzbekistan.
Moldova rarely makes the news and has amazing wines to boot.
But most of the African wildlife documentaries you see are filmed there, since that’s where the Serengeti is.
Actually, hordes that stole hoards.
Apparently, if you just overlook all of the brutality and mass murder, ol’ Genghis was a net positive. Dan Carlin’s been treating the topic at length in his most recent Hardcore History podcast series. (Mind you, he just mentions the distant, historical view, removed from the untold suffering that the Mongol conquest caused, as a jumping-off point to explore what this goddamn force of nature would’ve been like to experience on the ground, so don’t go there if you’re looking for the positive spin.)
My vote would probably be for one of the -stans, but Burundi might deserve an honorable mention. All I know about it is that if the President invites you in for coffee, it might not mean what you think it means.
I nominate you as our resident Mongolia expert.
IIRC it has very few paved roads either, a couple of miles or something kerazy like that.
For a brief time in the late 1970’s, the CAR called itself the Central African Empire and was all over the news. They had a guy who called himself the Emperor Bokassa I and was batshit insane even by African dictator standards. Since he was deposed in 1979, wow, it has been obscure.
Honestly that pretty much sums up all I know about the CAR. Well I also know it used to be a French colony and Bokassa was a canibal.
Too bad Delaware isn’t a country.
Paraguay.
I know nothing about it, beyond the fact that it had a little person as dictator, & a fought a very stupid war with its neighbors.
Democratic Republic of the Congo (formerly Zaire). A big country with a lot of people and problems but rarely heard from in world events:
Wikipedia: “It is the second largest country in Africa by area and the eleventh largest in the world. With a population of over 71 million, the Democratic Republic of the Congo is the nineteenth most populous nation in the world, the fourth most populous nation in Africa, as well as the most populous officially Francophone country.”
Constantly in the news in Australia, you know the whole Bali bombing thing and kids at schoolies.
I think we have a winner… though Indonesia arguably wins on a per capita basis
Honestly, if you think Brazil or Indonesia is in any sense obscure, that’s a statement about you.
I understand the sunny sky is aqua blue, and the couples dance cheek to cheek? Lots of pretty girls, and plenty time for good romance. 