Hi Peter,
I am not from Laminar Research; just a flight simmer from downunder. My comments and or advice is based on the fact you are running a genuine official version of X-Plane purchased from Laminar Research and not a steam version.
If you are running a steam version then you need to take your problem to a steam forum. This is not a steam forum but a forum for the Laminar version of X-Plane. Steam is a 3rd party hybrid that has been modified from the Laminar version. It is not the same product as the Laminar version. It installs differently and does not mix and match with the Laminar version. Laminar does not market steam. They direct you away from their site.
To the best of my knowledge there is no limit on the number of activations being performed on your computer with respect to the Laminar version.
Although you haven't stated what version you have ie Digital download or disc version, the software will perform silent background check to determine if you have a legal version or not. In your case because you are being asked for your product key or activation code I have determined you are running a digital version.
For X-Plane to run satisfactorily without asking for the product key as an ongoing annoying question you have to be connected to the internet at all times so that a silent background check can be carried out at any time by the software. If you don't, the software will ask for the key at intermittent periods (in your case 4 or 5 week) when you start X-Plane again. Alternatively it will determine that you have a demo version and will limit you to a 10 minute time interval at each startup.
From reading the subject heading I am of the opinion my response is quite clear. If judging by your opening words my comment is vague or does not answer your question then please be more specific of what your problem is.
Glenn