Thanks for the suggestions. Ultimately, while I didn't follow any of the patterns that were suggested, they helped me think of other cool adjectives for cultural levels as well as a more appropriate ordering. I've adapted my latest thinking into the following twelve non-zero culture levels.
NONE: 0
MINIMAL: 1-9
POOR: 10-49
FLEDGLING: 50-99
PROVINCIAL: 100-499
DEVELOPING: 500-999
ESTABLISHED: 1,000-4,999
NOTABLE: 5,000-9,999
REFINED: 10,000-24,999
SOPHISTICATED: 25,000-49,999 (still looking for a better name, but don't really love what I've seen so far)
INFLUENTIAL: 50,000-74,999
RENOWNED: 75,000-99,999
LEGENDARY: 100,000+
Note that the maximum expansion (ie, legendary) is now at 100k culture for a normal game. This is greater than 90,000 which Afforess was using, but I think it's probably still reasonable. Also note the discontinuity between the values under 10k (which increment in a 10->50->100->500 pattern) and those over 10k (which increment at multiples of 25k).
I'm going to preserve the same scaling ratios that Afforess used in his mod, so epic speed will require 200k culture to hit legendary, and snail needs a whopping 600k.
Obviously, adding two more levels means that the city will spread culture over a wider territory. Each city will now have a cultural range of 12 in any direction, which should be effectively 7-10 tiles from the city center when terrain and rivers are factored in. While that might be slightly more than I'd like, I think the cultural range math works out much better this way. Also, in the late game, this will mean that culture can really spread far from the city center, which can be highly destabilizing for small empires with little culture.
With this many cultural levels, though, I'm clearly going to have to play with Firaxis's internal formula for spreading culture to tiles. Firaxis's formula it increased the amount of culture spread to a tile by an order of magnitude each time the culture level increased. I can't have a city spewing 10^12 culture onto its own city tile each turn. That's the next thing I'm going to play with...as soon as I verify that the game loads correctly with these levels included.
If you know of any mods that will break when I add these additional culture levels, please let me know. For instance, I highly suspect that revolutions will be hammered by these changes.