To loop the current video, just right click on the player and select "loop".
This only works with public or unlisted playlists.
Any "unavailable" videos encountered in the playlist (e.g. ones blocked on copyright grounds) will be automatically skipped. Deleted videos will not be recognized.
Why is this Necessary?
For whatever reason, YouTube has a very poor playlist randomization player. I frequently listen to long music playlists on YouTube while I study and work and while doing so have run into many problems including repitition of songs, the player randomly exiting the playlist (and loading videos in the sidebar instead), and more. I was using this website as an alternative, but due to the popularity of the site (due to the ineptitude of YouTube :)), it was reaching its daily API request quota early morning every day, meaning it wouldn't be able to work at all past mid-morning daily. So, I got fed up and made own version.
A "loop" checkbox button (added next to the "mute" button)
Display the name of the current video just above the player