I used to have tabs that followed quite an in-depth structure, but I can't replicate this with 4.7.13.

Tab
-Tab
--Tab Item 1
--Tab Item 2
--Tab Item 3
--Tab Item 4

The drop-down option, I don't like. I don't think all visitors will realise you have to press on it.

I don't see any means of going beyond the depth of two links.

Am I missing something?

Im not actually sure what you mean here, to be honest. Are you looking at 3 levels of bar here?

Yes, three levels in depth. 

You have a parent tab, then a secondary underneath that but after it goes red. You can't have a depth of 3 levels, which means you can't expand on items much further.

There is no way in which to do this with the native platform. Nor has there ever been on version 4