Is it possible to grow shop bought garlic / onions / potatoes?
Yes, it is possible. I have Elephant Garlic that I am growing from cloves bought at the local market. I have also grown shallots bought at the market. A word of caution about the potatoes: If you use market-purchased potatoes for seed you risk bringing diseases into your garden. Using Certified Disease Free seed potatoes is a much better idea, although they are slightly more expensive. To me, the initial expense of certified seed potatoes is far less than the expense of correcting diseases brought into the garden to save a few pennies.
