I was quite surprised to discover there are now only 3 hard drive manufacturers (Western Digital, Seagate & Toshiba) who make every single magnetic hard drive in the world (down from 10 in 2000). Usually with most industries, once you move past the couple of big players, you can always find a couple of niche companies that fill up the rest of the market.
For example, x86 CPUs are dominated by Intel and AMD but there’s also quirky like Via in the corner, doing it’s own thing. Desktop graphics cards are dominated by nVidia, AMD & Intel but then there’s also S3 and Matrox who apparently are still making cards. Commercial airlines are dominated by Boeing and Airbus but there’s a bunch of smaller manufacturers like Embraer, Bombadier, Fokker, Iluyshin, Sukhoi etc.
What are other industries in which there is not a single small player in the market once you move past the few big players?
Large (>200 MW) gas turbines for power generation are all (as far as I know) made by 5 companies (1 US, 1 Japanese, 1 Swiss/French, 1 German and 1 Italian). But the Italian one is much smaller and manufactures under licence to one of the others, so it might qualify as a “niche player” per your criteria.
Looks like we’re down to four manufacturers of plasma panels. Panasonic is said to be abandoning that technology in March, leaving just NEC, Pioneer, LG and Samsung. Everyone else buys their panels from these four.
There’s quite a number of state-run or semi-public oil companies in the world, which may monopolise crude oil production in their countries and may be large in absolute terms but are not big enough on a global scale to be counted as one of the few really, really big players worldwide. Petrobras, Petronas or OMV are just a few examples. On top of that, I’m pretty sure there are privately owned oil companies active beyond the few big players such as Exxon or Royal Dutch Shell. Oil is surely not an example for what the OP is thinking of.
There are of course many very-small food product makers, but if you were to leave only the products made by the big three food congloms on the shelves… your local grocery store wouldn’t change much. If you reduced it to Kraft and FritoLay alone, you’d probably no more than wonder why the shelf stockers were being so lax.
Even in the US, you’ve got small private companies like Sinclair Oil (green dinosaur.) It owns a couple of refineries and a couple thousand miles of pipeline. It is mostly noticeable in the Northern Mountain States, but has retail operations in pretty much every state west of the Mississippi.
Well, in fact there’s a chance that it might say “Talon” on it. And I just looked at a couple pairs of my pants, they said “DLN” on them. From wiki:
Note that those three only add up to 60%, though I’ll concede that YKK’s share may have gotten larger since 2003. DLN turns out to be a trademark for zippers made by JoeLee Corporation, a South Korean company.