Is there a maximum number of encryptions I should make with the same RSA key?
Is there a maximum number of encryptions I should make with the same RSA key?


No, at least, none that can be practically performed.

For RSA, we assume that the adversary has the public key, and hence is able to generate his own plaintext/ciphertext pairs, and in fact, can do that a huge number of times, should he find that helpful.

Hence, if RSA becomes insecure if you encrypt, say, $2^{80}$ messages, then a large adversary can go ahead and encrypt that many messages himself (and thus obtain whatever weakness that was posited.

The fact that we believe that, say, 3k RSA cannot be broken with less than $O(2^{128})$ operations would therefore say that it's safe to encrypt $2^{128}$ messages (and you'll never ever need to encrypt that many).