Backtick! I came up with it on my own, but a few months ago I did some analysis of tmux.conf files on github (via the GCP public db) and it turns out that it's the 3rd most common prefix (after Ctrl+A and Ctrl+Space).
I have a nested setup with a "meta" screen session with escape key ^Z, which holds named "project" screen sessions with escape key ^O.
I background tasks in the shell seldom enough (because I'm always in screen) that I don't regret ^Z being slightly less convenient. ^O is a fairly unknown but very useful insert-mode binding in vim, but I use emacs these days so I don't notice it being escaped.
Before I used separate escape keys, it was maddening trying to remember how many times to hit ^A to send commands to which nested session.
I use Ctrl-J. So many C-* keys interfere with Emacs / readline.