The opponent possesses a string of ciphertext and has access to the encryption machinery as black box. What is the name of this attack?
Actually, this specific attack against RSA is known as a 'cycling attack'; it works because RSA defines a permutation, and so by doing repeated encryptions a sufficient number of times, you'll always get back to where to started with eventually.

However, people have analyzed this attack, and shown that (with extremely high probability) the amount of work required by this attack (for a randomly chosen RSA modulus) takes more work than the known factorization methods.