Steam badge didn't level up
Solution 1:
Is the game count on your profile 5 or are you counting the ones on game list?
Game count on profile counts some games less (for example F2P games aren't counted but some paid games aren't counted either, I don't know why they're not counted), and the badge is awarded based on the number on your profile.
If the game count on your profile is 5, then you have encountered a bug and you probably just need to wait a bit for steam servers to settle down a bit (it's sale season) or just buy another game and see if that works.