When I think of encryption and doubling it, I believe the strength of encryption, in the case of AES, as being tied to the number of rounds through the algorithm.
I don't believe that's the best way to think of things.
The strength of the encryption really is 'how much work does an attacker need to do to perform the attack'.
For a cipher, there are two strategies:
We can just go through all possible key values, and see which one 'works'
We can rely on cryptographical weaknesses within the cipher itself
Increasing the number of rounds can plausibly (not always, but in practice, appears to) make the cryptographical weaknesses within the cipher harder to exploit, but (unless increasing the number of rounds also increases the number of key bits), doesn't do anything to slow down brute force attacks (apart from a constant factor).
As far as we know, there are no known cryptographical weaknesses in
AES-256 (apart from some related key attacks, typically not applicable), and so adding more rounds doesn't increase security in any meaningful sense.
Now, it would look like running the text through two separately keyed AES-256 would double the number of key bits, and so significantly increase security. However, it turns out that there is a better-than-bruce force attack; this involves encrypting the plaintext with all possible AES-256 keys, decrypting the ciphertext with all possible AES-256 keys, and then searching for a match.
Of course, the amount of work involved is ridiculous; it will never be the case that anything involving $2^{256}$ operations will be feasible; this can be shown by thermodynamics and the minimal amount of energy needed to perform any operation.
This implies that, barring a cryptographical advance that allows us to break AES-256 with significantly less than $2^{256}$ operations, it'll be safe.
Which leads us directly to the next topic:
I have read some whitepapers on cascading encryption, but they referred to using two different ciphers, not the same cipher twice.
This potential cryptographical weakness is precisely why people use distinct ciphers. There might be a weakness in AES-256 that makes the cipher significantly weaker than our current understanding; however, if we perform AES-256 encryption, and then (say) Camellia-256 encryption, then the resulting cipher is strong unless both AES-256 and Camellia-256 are weak. It is possible that there is a weakness in one of the two ciphers; it is considered far less likely that both contain weaknesses.
