Analysis of Repeatedly Enciphered Plaintext using Same Algorithm / Key? Please forgive the impracticality of this question, but I'm curious about the behaviors of encryption algorithms applied to their own output.

Suppose I have an encryption algorithm E and (using the same key(s)) I repeatedly encrypt its output for a given plaintext: E(E(E(E(...E(plaintext)

Would a modern encryption algorithm, such as some variant of AES, display any kind of cycling with respect to its output for arbitrarily long compositions of itself?  Is there some known number of compositions where the final output is itself the plaintext?
Eventually, it will cycle, but it'll almost certainly take far longer than your lifetime to do so.

If that sounds too flip, here's the mathematics.  Assuming AES is secure, then the probability it will revisit the original plaintext after at most r repeated encryptions is just r/2128.  (This follows from a standard fact about the cycle structure of a random permutation.)  For the values of r that can realistically arise in practice, this probability is negligible: less than the chances of getting hit by lightning, twice.

Is there some known number r such that r repeated encryptions are guaranteed to bring you back to the plaintext you started with?  Yes, for instance: r = (2128)!, i.e., 2128 factorial.  Or, r =  lcm(1, 2, 3, ..., 2128) also works.   However, this is a stupendously, ridiculously large number.  So it's completely irrelevant in practice. 

Bottom line. If you've got a secure block cipher, you don't need to worry about cycling.  It's not an issue.  It just ain't gonna happen.