What are the medical benefits to breastfeeding past a year?

Plenty. Breastmilk changes in composition as your child grows. The infection fighting properties of breastmilk are still present after a year; lysozyme, lactoferrin and secretory IgA are present in stable amounts. Plus we know that moms who nurse longer lower their risk of breast cancer. Human milk is specific for the needs of human babies. There's no cow out there making better milk than you are!
Small added benefits. In the United States, a normal healthy baby who is growing well, will not see much added medical benefit from breastfeeding beyond age 1 year, compared to if he were to stop breastfeeding and just eat a variety of solid foods and drink milk or formula. The antibodies, white blood cells, iron, and miscellaneous chemicals in the milk are unlikely to make a visible difference in the toddler.
Security, comfort. There aren't many medical benefits to breastfeeding past a year, but there are a lot of social benefits to both child and mother. The child feels secure and comforted while breastfeeding and it continues to strengthen the growing bond between mother and child. They will still grow up to be independent thinkers research has shown that a supportive home environment early leads to independence later.
Immunity and bonding. After a year there continues to be a boost to the immune system and kids show fewer infections while breastfeeding. Mothers can also comfort their toddlers with breastfeeding if they are injured. This does not prevent kids from becoming independent. They tend to be very secure in themselves later on! mom's risk of breast cancer also decreases with prolonged breastfeeding.