Puzzle Rings with Chains
There are some puzzle ring variants where a single 'band' goes around the finger multiple times. These have not to my knowledge been studied thus far. The best way to actually make them would be for the band which goes around multiple times to be made as a chain, with universal joints connecting the segments.

Here's a simple but pleasing example, with a single chain which loops around three times:

______   ___   ___
      \ /   \ /
       /     \
___   / \   / \___
   \ /   \ /
    \     /
___/ \___/ \______

Here's a slightly trickier example with two traditional bands and one chain. The principle behind this one is very interesting.

______   ___   ______
      \ /   \ /
       \     \
___   / \   / \   ___
   \ /   \ /   \ /
    \     /     \
___/ \   / \   / \___
      \ /   \ /
       \     \
______/ \___/ \______

The following puzzle consists of two chains which both go around twice, and is a bit of a departure because they can be completely separated when taken off. The two chains form a loop knot around each other, sort of, and the puzzle as a whole has a subtle asymmetry. It might be very tricky.

_________   ___   ___
         \ /   \ /
          /     /
______   / \   / \___
      \ /   \ /
       /     \
___   / \   / \______
   \ /   \ /
    \     \
___/ \___/ \_________