South Georgia has no nationals and no permanent residents. I also noticed a couple of pacific islands included that do not have electricity. British Virgin Islands is an important relay station, there are always techies there.
Rarest passport stamps

I have Vatican City, Zimbabwe and zambia in my passports, and I didn't know it until I checked my passports and went down. I was dumbfounded when I saw them.

North Korea should theoretically be the rarest one, but I think people realized that and started putting it as their country as a joke, so it went up. In fact, I think I've had 4 or 5 "North Korean" opponents in my time.
If we were to suddenly bring attention to the other ones that are almost impossible as well (see the above replies, for example), I'm sure they wouldn't be so "rare" either after a while.

In general, all the Island nations are rare, many of the Caribbean Islands were my last to get and I am still missing 2, Monserrat and Anguilla. Additioanally, I am missint the island nations of Northern Mariana Islands and Niue. The only other 2 I am missing are Olympic and Paralympic. When did they add these, this is the first time seeing these.

Bonaire, Sint Eustatius and Saba
Bouvet Island
British Indian Ocean Territory
Christmas Island
Cocos (Keeling) Islands
Comoros Islands
Cook Islands
Dominica
Equatorial Guinea
Falkland Islands
French Guiana
French Southern Territories
Grenada
Guinea
Lesotho
Liberia
Mali
Mayotte
Netherlands Antilles
Norfolk Island
Northern Mariana Islands
Pitcairn
Saint Barthélemy
Saint Helena, Ascension and Tristan da Cunha
Saint Lucia
Saint Martin
San Marino
Sierra Leone
Solomon Islands
South Georgia and the South Sandwich Islands
Svalbard and Jan Mayen
Timor-Leste
Tokelau
Wallis and Futuna
these are some countries that i have never played according to chessmonitor.com
I think Zimbabwe is the rarest in my opinion! I got one of those. I think you just get the passport after a player match when I think the player doesn't live from that place. Cuz people can change their flags
I have Zimbabwe stamp