As an isolated example of this kind of thing, consider the shape of the Earth as discussed in Asimov's "Relativity of Wrong" [http://chem.tufts.edu/answersinscience/relativityofwrong.htm]

You are both right. Animal reproduction is exponential as long as resources are freely available -- the more animals you have, the more they can mate. Like the parent post, the greater the perimeter of your knowledge, the greater the boundary you can now explore from.

Once animals hit the resource limit of the environment, the exponential curve flattens out. Like you note, the more of the map you cover, the less there is left to explore.

The interesting question is how big is the map and how close are we to reaching it's boundaries?

Personally, I think it is effectively infinitely large. If you consider "discovery" to include new combinations of existing things, then you're talking about permutations. If our universe consisted only of 52 playing cards, there would still be 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 different ways for us to discover that they can be shuffled together.

Ok, now postulate that we can only invent things on the border of that map (because of technological limitations). Then you'll get a very complex dynamics, of the invention possibilities increasing and decreasing, some times fast, and changing almost unpredictably.

If the illuminated areas are a very small portion of the entire map, then the circle metaphor actually may be more appropriate. But I see your point.

Hmm... Perhaps better still would be an immense valley of very uneven steepness and roughness, that we are climbing out of.

Does the set of everything exist? Why or why not?