We bought a place in central Maine last year with an existing kitchen
garden that seems to lend itself very nicely to a "Foursquare" layout. I
would like to pursue this, but I'm confused about a few things.

It seems like a foursquare layout will give you raised beds with depths
much larger than the 4 feet I've always heard you wanted for raised
beds. The 4 foot depth is to enable you to reach into the garden from
either side and never have to get up and walk around in there. Those are
among the benefits of raised beds.

What am I missing here?