Loop video on iPhone - Settings, Photos, "REPEAT"
Yes, iOS 7 gives you this option for free.
Go to Settings.
Go to Photos & Camera.
Scroll down to "SLIDESHOW". Turn on "Repeat".
Create a new album, put the video(s) you want to loop into it.
When you want to play it, select the album.
Select the video.
At the bottom of the screen is the little "box with an upward arrow" icon.
Press that icon.
You get a screen that allows you to select "Slideshow". Press "Slideshow".
You get a screen that allows you to select iPhone or AirPlay. Make your selection.
Press "Start Slideshow"
It loops.