Onions
Onions will often soften and even turn moldy if stored in the refrigerator. They last longest when kept on the counter. Onions need some air circulation and can be stored in the mesh bag they are often bought in.
Garlic
Garlic is a food that should definitely not be kept cold. Garlic can harden in the fridge and start to grow mould. It can even start to sprout. Garlic is another food that benefits from circulating air. It will last well for over a month in the open on the counter in a basket.