Steam and recently played games jump list
Solution 1:
This is basically an UI limitation with the jump lists. Unfortunately they're limited to 10 entries per program. By default, Steam provides several entries such as "Shop", "Friends", "Quit Steam", etc. that will eat up that number.
In my case, I've only had space for 2 recent (or pinned) games.
To solve this, you'll have to remove some of the default Steam entries that you don't use anyway.
Go to Settings, Interface, and then find the button Taskbar Preferences on the bottom. Click it, unselect all entries you don't use in the jump list anyway, and you'll once again get one or more recent games entries.