i do this, and while some things (tender greens, grapes) won't quite make it, there are enough good fresh fruits and veggies to make it work. i mean, no one will perish if they have to go 2 or 3 days without baby spinach, right?
if you buy bananas very green and separate from the bunch as soon as you get home, they ripen more slowly. apples and pears are fine for 2 weeks, and most berries and succulent fruits will go over a week in the fridge (and freeze them when they start to get close and use them for smoothies.) winter squash keep forever, and zukes, cukes and tomatoes are generally fine for 1-2 weeks in the fridge if you buy 'em fresh.
the only thing i buy every week is milk, because i can't stand processed milk, and raw doesn't keep as long.