Given one message encrypted repeatedly with the same 'n' but different 'e's, is it possible to decrypt the ciphertext without simply bruteforcing n? I'm trying to solve a problem where I've been provided with a few public keys and the ciphertext, and I've been told that the plaintext was encrypted repeatedly using all of these keys. However, all of the keys have the same $n$ and a multitude of $e$s, varying from 2 to 126 digits. All of the other attacks involving matching $n$s only seem to work for multiple plaintexts encrypted with multiple keys, where in this case it's one plaintext encrypted with multiple keys. Is there any attack other than brute factoring that could allow me to find the plaintext?

As a better example:
Bob is concerned about his security, so he asks Alice to produce 3 keypairs, $(n, e_1), (n,e_2), (n,e_3)$ and then encrypts his plaintext, $M$, with each of the 3 keys, feeding the ciphertext $C$ from each into the next, so that by the time he's done, $M$ has been encrypted with all three keys in series to produce one final $C$ which is sent to Alice.
Ok, here's a hint:

Suppose you know the values of $P^{e_1} \bmod N$ and $P^{e_2} \bmod N$.  How could you use those two values to compute $P^{e_1 - e_2} \bmod N$?  How could you then use this repeatedly to compute $P^{\gcd(e_1, e_2)} \bmod N$?