I have noticed that old iTunes version may repeatedly ask for your Apple Account password when launching iTunes. You provide it, and the dialog annoyingly and repeatedly reappears. If that describes what you are experiencing, I suggest typing that password somewhere (locally, on your Mac, such as in the TextEdit app) cutting it, then using ⌘ V to paste it in the password field as many times as required to placate the nag. Many repeated efforts may be required. Quit the TextEdit app when you are finished (obviously) and in an overabundance of caution, copy something else to overwrite the Clipboard's previous contents first.
- You should not normally type, write down or otherwise record passwords in any readable form anywhere, ever.
I suspect that iTunes is requesting authorization for each specific iTunes Store purchase you may have made using an Apple Account password that may have changed since then, but that is little more than a guess.
Just make 100% certain your Apple Account password is correct. If you repeatedly enter incorrect Apple Account credentials your digital life will become more challenging than usual... temporarily.