Is it obligatory for me to convey greetings from a very forgetful person every time he repeats his request, or is it sufficient to convey it once? And how can I refuse to bear the responsibility of conveying greetings without being obliged to do so?

The more accurate view is that you are not obligated to convey the greeting (salaam) unless you have undertaken to do so. If you committed to conveying it, then you must convey it; otherwise, you are not obliged. So, if your words did not imply that you bore the trust of conveying the greeting, then you are not bound to convey it.