Watches.
My mother-in-law has a friend whose husband invented the Botsball thermometer and started a company that made only that product for a few decades. When he died his wife continued the company until she retired, presumably selling the design to a larger company. The company was successful enough to provide a comfortable living and worry-free retirement.
Then again, the Botsball came in Fahrenheit and Celsius versions, so I guess even that is disqualified.
Sriracha.
Victorinox also makes a line of kitchen knives. Excellent ones, I might add.
I suspect that many companies started out with one product, but I cannot imagine them not diversifying if successful. If not, they folded their tents and slunk off and we’ve likely never heard of them. I imagine some early bicycle makers did just that. They were successful and then branched out.
Each of those comes with or without magnetic shielding.
Diversification! 
+1
I was going to post Legos but damn, De Beers is a world juggernaut …
This is really the question the OP has to address. Otherwise, this thread is pointless, even if can manage to define “single unique product.”
Lena Blackburne’s Original Baseball Rubbing Mud, the mud they… well… rub onto baseballs, is the only product the company makes. They also sell T-shirts and some other memorabilia, but somebody else makes those.
Lego makes more than just lego bricks, but even if it didn’t, it makes them in different sizes and colors. From what I can tell, that would disqualify it according to the OP. No doubt DeBeers would be disqualified because the diamonds come in different weights and colors.
The Brannock Device Co, Inc. has to be considered. The only make one thing: the Brannock Device, something everyone here (if they live in the US; I’m not sure if it’s used elsewhere) has definitely used at some point (though I doubt anyone here has ever bought one). They’ve been around since 1927 and going strong.
Their web page only has Brannock devices, though they do come in different sizes and variation. But they make nothing else. They had sales of $1.5 million in 2001, despite the fact that they never wear out.
Not really, because you haven’t defined “successful.”
People are just posting whatever single-product company someone happens to know about, without providing any real objective criteria for why it should be “most successful.”
“Textual Assault” seems to pretty much come out with the same thing every time.
I gather this refers to Huy Fong Foods.
In a similar vein, the McIlhenny Co. of Avery Island, LA for a long time was known only for Tobasco (these days, they’ve branched out into a few other hot sauces, including a version of sriracha. 
DeLorean automobile before the drugs?
Any professional sports teams qualify? They sell only entertainment (and angst.)
Nitpick: Tabasco.
And yes, they’ve added some nice variants to their line. Frankly, their original sauce was never one of my faves, but I love their green sauce on raw oysters, and their chipotle is quite fine on mac and cheese, or my enchiladas (but not as good as cholula, IMHO).
I just picked up their version of sriracha, looking forward to trying it.
The normal meaning of the first sentence would be that the product is unique to that company. Otherwise you could just say ‘to only produce a single product’.
Reading it the way it’s written GM doesn’t fit mainly because loads of other companies make vehicles more or less equivalent to GM’s, not just that GM itself makes more than one type of vehicle.
And it would suggest as a real answer Polaroid. That was at one time a highly successful and fairly large for its time company that made a basically unique product, instant cameras, which were largely its only product though when past its peak it tried its hand at a number of other things. And as everyone knows, the company eventually went bankrupt, but even successful companies usually disappear eventually.
The strictest way of interpreting the OP’s criteria would be “a company where, if you buy their product, all you have to specify is the amount. No variations whatsoever.”
As it happens, I do know one company which fits that bill, the American Doughnut Kitchen in the center of Melbourne. Not as large as WD-40 or the Brannock Device company, but it has been going for 68 years selling precisely and only one thing - a jam filled sugar coated doughnut, no options, no variations, no special orders. You rock up to the counter and say “two please” and get two of the product that they make, and that’s it.
They do not appear to have any plans to diversify.
Nobody would buy any of their accessories unless they also had their 1 and only main product. So they are add-ons, not devices in and of themselves per se.
Outside of a kids lemonade stand I don’t think you are going to find any business that meets your screwball specs. A shoe company that only sold 1 specific pair of shoes in only 1 size wouldn’t fit your specs if they sold replacement laces, would they?
As has already been mentioned most successful business eventually branch out even if the other products are similar to their original.
I think the apparent extreme strictness of the topic is generally a less profitable way of doing business, so it isn’t usually done. I do think there are numerous companies with a flagship product that was their first big success and still is the majority of their business.
Steinway New York still makes at least Models B and D, probably others too…
A lot of the trades are just one product. If I call up a plumbing company all I’m going to get is plumbers.