I would like to store encrypted AES-256 keys in a public location and give 100 different people access to the AES key by using each of those 100 people's public RSA encryption key to encrypt the AES key. How much does this degrade the encryption? The AES key would be random, not passphrase based.
Assuming the RSA key was of sufficient size, it would not degrade the encryption meaningfully. RSA does not lend itself easily to batch attacks. Of course, there is still the concern that those 100 different people all have access to the key, so all it takes is one person to be compromised to compromise the other 99. But from a purely cryptographic standpoint, you will not be harming security.
If you want the nitty-gritty math behind it, you may want to ask on Cryptography.