I see the search box in the upper-right of every page on every SE site has a
tabindex of 1. Which is really useful and handy as it is the most used by users.
I suggest, to add a
tabindex of 2 to the
StackExchange link in the top-left corner of the window clicking on which pops a Notification box with options
Screenshot, to be more specific:
This would be helpful, and would save a little user key-board homing time.