I was only able to go 1 level deep. I could open the top level browser and load the site which allowed me to open another browser but that wouldn't load the site.

Properly coded, there is no reason to assume, that it would not work for an arbitrary depth. Considering, that this is on a website, and that it is running JS, which in most implementations does not implement tail call elimination, coding it properly probably involves externalizing the stack.

I tried on my phone, and found that I ran out of screen space before it stopped working.

It was interesting, seeing all of the clocks tick, one above another, second after second.