To what extent are the naked eye stars classified?

If I went to the middle of the Sahara, or the top of a mountain, basically the best dark sky site imaginable, how many of those stars I see with my naked eye have been categorized? By categorized, I mean having a name or designation, or being assigned a magnitude in some official record.

If the answer is “all naked eye stars”, then how much aperture would I need before the number of stars that are visible exceeds the documentation of scientist? Are all stars visible with 10x50mm binoculars documented etc? At what point are the stars so numerous and similar that naming them becomes pointless?

Some links to get you started
http://www.nightskyatlas.com/starnames.jsp
List of proper names of stars - Wikipedia
http://www.astro.wisc.edu/~dolan/constellations/starname_list.html

I’m pretty sure that every star that can be imaged has been or is in the process of being given a designation, for the simple reason that you can’t do astronomy if you don’t know what you’re looking at and what’s new and different.

The Wikipedia page on star catalogues is a look at historical atlases. The largest one seems to be:

Amazing! This far exceeds what I expected the answer to be!

AFAIK those star catalogs are formed by using software to scan and analyse extremely high resolution photographic plates. Then the software assigns numbers to each star in a systematic way depending on its plate number, position and magnitude. The software can apparently distinguish between stars and galaxies and binary stars.

So its not like humans assigned all those 945 million star numbers by hand.

Not to mention that most of them are not visible to the naked eye as required by the OP.

To answer the second part of the question, If all the magnitude 21 stars have been cataloged, that means you’ll need a telescope big enough to see 22nd magnitude stars. I think that puts you somewhere between 40" and 60" of aperture - in the range of a mid-sized scientific observatory.