Is there a connection between uncountable sets and exponential growth? Let $S_n$ be the collection of all binary strings of length $n$.  It seems that as $n$ goes to infinity, $S_n$ becomes the set of all infinite binary strings.  Each time we increment $n$, the size of $S_n$ doubles, so $|S_n|$ is undergoing "exponential growth" with respect to $n$.

Similarly, let $C_0$ be the set $\{0, 1\}$, let $C_1$ be the set $\{0, \frac{1}{3}, \frac{2}{3}, 1\}$, let $C_3$ be the set $\{0, \frac{1}{9}, \frac{2}{9}, \frac{1}{3}, \frac{2}{3}, \frac{7}{9}, \frac{8}{9}, 1\}$, and so on.  It seems that as $n$ goes to infinity, $C_n$ becomes the Cantor set.  Again, each time we increment $n$, the size of $C_n$ doubles, so $|C_n|$ is undergoing "exponential growth" with respect to $n$.

In both of these cases, the "limiting result" of "exponential growth" is an uncountable set.

In contrast, let $K_n$ be the set of $k$-tuples of $\{1, 2, ..., n\}$.  The size of $K_n$ only undergoes "polynomial growth", and in this case, the "limiting result" (set of all $k$-tuples of positive integers) is a countable set.


  Is there a connection between uncountable sets and exponential growth?


In particular, I'm wondering if there any interesting results that generalize or extend the observations I made above.
You are mistaken in your first assessment. The limit case is countable.

The limit case of $S_n$ is not $2^\omega$, the set of infinite binary strings, but rather $2^{&lt;\omega}$, the set of finite strings of arbitrary length. Similarly, $C_n$'s don't limit as the Cantor set, but rather as the set of endpoints of removed intervals, which is just a countable set.

The issue here is the same in all cases, $$\sup n^k\neq n^{\sup k}.$$
Here is an example of a connection between uncountable sets and exponential growth. Consider a binary tree of infinite depth.  Each path down the tree codes a binary string of infinite length, and all possible binary strings of infinite length are accounted for by the paths in the tree.  The Cantor Diagonal Argument shows that the set of all possible paths cannot be put into a 1-to-1 correspondence with the Natural Numbers, in other words not countable.  The branch/node count grows exponentially as one moves down the tree (it doubles at each subsequent level of the tree.)