How many religions seeks enlightenment through reincarnation?

3 certainly do. Buddhism - which has the central dogma of continual reincarnation on the search for enlightenment. Jainism also seeks enlightenment through reincarnation, which holds that all life is sacred and that life is cyclical. Hence adherents of Jainism eat no meat and avoid violence to any animals (insects included) as they may be reincarnated as an animal. Hindus are also firm proponents of enlightenment through reincarnation.