Any vegetable that has a base can also be regrown such as Bok Choi, Napa cabbage, etc. You won't often get the same sort of veggie that you bought, often just some new leaves, but those are great and best of all they're free. I've seen second growth Napa cabbage in the Korean market before. After the harvest of the original cabbage, the root is left in the ground. A rosette of new leaves forms and the plant is then dug up, the roots removed and the rosette sold. Haven't tried it but it must be some sort of delicacy.
I've wondered about growing garlic in the tropics. It's always available, but garlic doesn't usually send out roots when the weather is warm. Up north I plant in the fall. It can freeze solid and still grow in the spring to be harvested in August, down south in June. Never tried to sprout it in water.
No wonder the roots are taking so long to come out...must be.