If we consider information to be safe if we encrypt it (e.g., text in a file, encrypted with modern strong encryption), would it be safer (as in harder to crack) if we then encrypted the encrypted file, and encrypted the encrypted^2 file, etc.? Is this what strong encryption already does behind the scenes?
If you use different keys, yes. That’s for example how it’s recommended to use post-quantum algorithms, you encrypt with them and a classical one.
If you repeat keys, it’s very complicated and probably not safe at all.