OK, so this happened to me, not just once, but a few times. I had an old PC, 2 old laptops, maybe 3, that became defunct and I either gave them away or ditched them depending on what state they were in. But I forgot to deauthorize my computers in iTunes. Urgh, and now I have hit the 5 computer limit of the number of machines that iTunes will let me play my songs on.
If this is you, and maybe you have already hit 5 and can’t authorize your new machine, then do not fear, there is a way round this. I discovered this when I set up my latest new PC and wanted to load iTunes on there. Luckily it was the 5th machine that I had authorized but I knew that only one of the other machines I still had in my possession so I also knew that there would come a point where I would likely want to authorize another new machine and I wanted to preempt that.
So basically what you need to do is to go to your account info page and you can deauthorize all of your computers – yes it is all or nothing! Then you can reauthorize the ones you are actually using.
You just need to enter your apple id and password to get to the account settings page so make sure you know this info.
Then you will get the option to deauthorize as shown below. Now you don’t see the actual deauthorize option on my screen shot below (that is why I have added it manually in the place you would normally see it) – because you are only allowed to do this once a year (at least I assume that is why you don’t see the button here – it may also be because I only have one PC authorized right now and I can do that from the store menu, so that may be it.