I do most of my shopping at the large Kroger in Green Acres Plaza, so most of my experiences are there.
Our local Krogers double coupons up to a dollar. That means a $1 coupon is worth $2. But they will only double if the coupon does not say "Do Not Double" (DND.) The store used to double coupons with the verbiage anyway, as long as the coupon barcode started with a 5. (Barcodes that start with 9 are made not to double, and are usually found in places like the blinkies in the store.) You may still be able to have some coupons double if they start with a 5 but have the DND, but I think the cashiers may have been informed to manually override the double. That's what I have seen anyway. Or maybe I've just been stuck with a very coupon unfriendly cashier. A lot of the coupons from coupons.com and various other internet printable sites have DND on them, but most coupons from the Sunday paper inserts do not.
The policy posted in the store is that they will only double 3 like coupons, and also they will only accept 3 like coupons.
It is also posted in the store that they will only accept internet printables up to $3. I'm not sure if that is only in my store. Kroger does not have a corporate policy on coupons, so your experience may vary from store to store, or even based on your cashier. Coupons.com does have a page here on Kroger's acceptance policy on internet coupons.
Overall, Kroger is a great place to shop as a couponer. Non-sale items tend to be pricier than other grocery or supercenter stores in the area, but if you track sales and match with coupons, Kroger is probably the best store in the area to score great deals.