Perennial

Perennial Vegetable gardens are the easiest to maintain as they require little or no maintanance on your part and will produce food year after year.
The downside of it is that these gardens won't produce as much food compared to row gardens.

Asparagus plants are probably the most common as they will grow back year after year with no work on your part you just need to plant them someplace where they won't get ran over by a lawn mower and thats it after that you can cut them when they mature and don't have to worry about replanting because its all in the root system.
Finding the right plants is not always easy as the stores like to sell people Asparagus roots that wont grow back the next year by selling you only "Male" plants the reasoning behind this is that they believe most people will see them as a weed but I personally think its because they want you to have to come back every year and buy more from them so make sure to ask your supplier if what they are selling you is only the Male variety or not.

Another popular perennial is Onions.
All you have to do is plant them someplace and instead of harvesting them all you simpley let some sit there and they will grow back next year eventually growing more onion plants in that spot so making say a large Mound Garden and planting it with onion sets will produce onions for you through the summer so when you need an onion or two for dinner you can just go to your onion patch and pluck one or two from it.
Unlike Asparagus you must weed your onion patch from time to time but I dont recommend taking out all the weeds as it will slowly reduce the quality of the soil just make sure the onions dont get overtaken by weeds.