I have upgraded from Lightroom 4 to 6 due to new camera (Olympus OM5mk2). Upgrade failed after a while saying free trial ended (I went down the purchase route only). I re-intalled and after a few weeks program would not load again beyond the box with the trees. This has now happened several times and each time I have had to re-install.
Are you Signing in and entering a serial number and are you staying Signed In?
Sounds like you never signed in and enter a serial number. Once you sign in and enter a valid serial number, and Don't Sign Out, LR should work forever.
Could also be a permissions problem that the license file isn't getting written to the hard drive because your username doesn't have Write permissions to the folder the license file is stored in.