These are quite rare. Let’s rule out initialisms or acronyms like AAA and KKK.
So far, all I have found are:
In Gã (language spoken in Accra, Ghana), the future tense prefix is aaa-. An example I found in M.E. Kropp Dakubu’s 1973 Ga-English Dictionary: Kofi aaaba (‘Kofi will come’). I picture the Ghanaian delegation at the U.N. saying this, expecting a visit from their old friend the Secretary-General.
French créée (‘created’, feminine gender). This is the one that made it into the Guinness Book of Records.
All those German words compounded of a word ending with -ß (-ss) and a word beginning with s-. Example: Faßstab (‘barrel stave’), which could also be spelled Fassstab. But I’ve heard this triple s was eliminated in the recent German spelling reform.
On the island of Oahu, there’s a town by the name of Kaaawa. When my family and I went to Hawaii, I made a point of going to that town and taking a picture of a sign for it (as well as for Aiea, the only city in the US without a consonant in its name).
That site you linked to ruadh, wouldn’t you know it, every time I get a nifty idea for a linguistic curiosae project, someone else has already done it better. Most of those triple-letter words are German compounds, as you’d expect. I had it backwards—the new spelling reform allows them now!
Scarlett, I’m sure you’ll be delighted to know that uncopyrightable is not the only word in the English language that can be spelled (properly) without repeating a letter.
Dermatoglyphics also has that characteristic. Granted it’s a plural, but that still fits.