I could be missing something, but it seems like none of the properties want to loop themselves. I may not understand the math that is being used behind the scenes for that setting, though.
However, I do have two potential solutions for you.
1) Set up a looping fractal in a precomp and use the Displacement Map with the Fractal as the displace map.
2) Use Turbulent Displace and turn on Cycle Evolution. Then, every 360º you will get a loop— or you can randomize it a little more by changing the Cycle (in Revolutions) number.
Neither of these will keep the line widths the same as if you were using Wiggle Paths, though.
One final solution that may not work as well as you would like:
Create a precomp with your shape layer that is half the length you would need and set the Wiggle Paths like you want it. Then bring it into your final comp and do a Time Remap on the layer. Extend the end of the layer to the end of your timeline and copy the first keyframe to the last frame plus 1 of the timeline. This will play the wiggle forward, then reverse it at the halfway point, and play back to the beginning. This will loop it!
Your screenshot only shows the cycle one per second. That's not very fast. This animator should run for as long as the lawyer lasts. Give us some more details on what you are doing. It works fine for me every time I've tried.
I thought to the same solutions, but what I need is a proper loop of the wiggle path. For now the best solution that I've found for my situation is to use some keyframes on the size of the wiggle to start and end at 0.
Anyway thanks for your help, appreciated!
I'm using a shape layer as particle with Time sampling set to start at random and loop. So without loop I will see some particles jumping from a shape to another one.
This is my looped particle using keyframes on the size of the wiggle to make it start and end at zero. The loop point is still clear and I'd like to have a proper loop of the wiggle to hide it: