I'm growing garlic in my garden this year, and was wondering if it turns woody (undesirable for cooking) if you let it grow for too long.
This is more for gardening SE but as I've been growing garlic for years I'll give you an answer. As it involves the major types and differences it'll be of interest to anyone interested in using garlic.
Garlic will not get woody. Bushy herbs like sage get woody, where the growth gets old and inflexible and the leaves don't have good flavor. All you need to do is prune the woody limbs aggressively and it will grow new shoots. If you let garlic grow too long what will happen is that the outer layers of skin will deteriorate and then the bulb will separate, here's a link to what that looks like. If you lose the outer layers of skin your garlic has a much more limited shelf life, and even less so if the bulb splits. You can still use it, but you'll need to use it soon after curing.
There are 2 major types of garlic: hardneck and softneck. Hardneck garlic has bigger bulbs with fewer larger cloves and tends to be stronger in flavor. Softneck garlic bulbs tend to be smaller and have more numerous and smaller cloves. Softneck tends to store longer than hardneck, although my garlic from both varieties lasts me a whole year.
The two types are very different when it comes to knowing when to harvest. Hardnecks send up a flower bud about 1-2 months before they are fully grown, you pinch these off as soon as you see them (use them in stir fries, they are delicious - and strong!) as if you let them grow into flowers they take much of the growing power away from the bulbs. Hardneck stems stay strong throughout the growing season and you can tell when they need to be harvested when about 1/3 of the foliage on the stem dies. If you leave it any longer you'll lose the bulb's protection and it could split. If you lift too soon your bulbs will be smaller than they should be.
Softnecks do not send up a shoot, and you can tell they are ready to harvest when the stems get soft and lay flat on the ground. This is known as "going weak at the knees". You'll want to lift them very soon after this or they could rot in the ground. When harvesting any kind of garlic you want to get deep under the bulbs with a shovel and lift them up from underneath, if you try and pull them up by the stem you will just rip the outer layers and ruin your bulbs.