These variants on the basic loop in loop chain are all related and simple in theory, so I dumped them all onto one page. Here you will find double and triple loop in loop, and single, double, and triple through-two variants. The techniques involved are basically the same used for single loop in loop, so if you don't know about that weave, you're not going to gain as much from this page.

Okay, first is double loop in loop. Try using 20 gauge 1/2" ID rings for this chain. Start out by making two "starter" rings and soldering them together as shown. They don't have to be soldered together for the next steps to work, but it does make things much simpler.

Add a new link through the first of the previous two links.

And now a new link through the second of the two.

Just continue this process to lengthen the weave.

The other basic variant is the tripled loop and loop chain. It's not that much different from doubled loop in loop in the theory. The practice isn't that bad either. I'd suggest 20 gauge 5/8" ID rings for this weave. Start by soldering together three links as shown.

Now add a link through the first one.

And the second.

And the third.

Continue to lengthen the chain. This one tends to be very thick, and it looks incredibly intricate.

The only other "simple" variant is the through-two chain. This chain is achieved by sending each new link through two of the previous links (hence the name). Obviously you will need somewhat larger links for this chain. I've used 20 gauge 5/8" ID with some success on the singled version.
The chain starts out like a normal single loop in loop chain.

Now, in adding the third link, pass it through both of the previous links.

Each new link should pass through two of the previous. Not much to it, eh?

You can also do this with double and triple loop in loop. Try 22 gauge 7/16" ID rings for double, 22 gauge 5/8" ID for the triple.

As a final suggestion, your links will probably not be perfectly consistent, despite your best efforts. You can use a draw plate to narrow down the chain, improving your link consistency and the flow of the chain. However, drawing the chain will work-harden the metal, so you may have to anneal it after every few passes depending on how much work the draw plate has to do. You can theoretically use a draw plate on any chain; it works best for doubled and tripled chains.