There is a difference to me between "great city state" and "fun city state".
There are many great city states - Bologna and Kumasi for instance, among the two best city states but hardly my idea of "fun" (it's a passive bonus that you rake in, and you dont really change your playstyle with them).
Fun city states however, that's more difficult.
I personally enjoy Nazca a lot. It's not fantastic, but it does give an incentive to settle more desert'ish areas.
It also requires you to think ahead in order to get good value out of the Nazca lines improvement.
It also helps that you actually get to keep the improvements afterwards, regardless of suzerain status, and the improvements themselves are gorgeous as far as tile improvements go.
Definitely a city state I enjoy seeing whenever there is some desert around, even if the game impact is usually quite minimal.
I also like Nan Madol and Anshan a lot.
Nan Madol is not only potentially very powerful (especially on water heavy maps), but it really rewards settling coast/lakes while it frontloads a resource that is usually quite difficult to get a lot of.
This unlike Auckland, which I personally believe is quite a boring city state and rather overrated.
Production is usually not too hard to get as a yield compared to culture, and Auckland doesnt frontload the yields the same way Nan Madol does (you need population to work all the tiles, and the sea tile needs to be better than any of the land tiles you have).
Nan Madol can often take care of all my culture needs, so that I don't have to bother with theatre squares or similar.
Anshan is just very handy in that it gives a yield (science) to a playstyle that usually involves sacrificing science for culture or faith (since campuses compete with theatre squares and holy sites for district slots), and thus allows you to skip on more campuses in favour of theatre squares and/or holy sites.
I've had a game before where I planned on playing for science, but changed my gameplan around when I spotted Anshan as my neighbour and had my great prophet available a few turns later, and instead went for a cultural reliquaries game with extra great works on top.
Not many city states will be able to influence my playstyle that much, but these (on the top of my head) can.