I saw this tip for a continuing supply of free basil in a newspaper and was amazed to find a) it worked for me and b) how fast it was, compared with growing.plants from seed.
Take one supermarket pot of basil, cut off a stem or two at the base and put in a glass of water. Within days there’ll be a mass of tiny roots. Pop the cuttings in compost and they will become new plants, ready to take over when the original one is finished. When these start to get too tall, pinch out the tops and they'll produce sideshoots and become bushy. Keep on taking cuttings whenever the plants look like flagging and you’ll have free basil constantly at hand on your windowsill, ready to toss into salads and sauces.
Here you can see the original plant (back) with the rooted cuttings and a new plant. Magic!