Behold, the Seemingly Immortal Hydra
Look in a freshwater pond and you'll find the hydra, a creature with a surprising superpower: it regenerates, and you can make two hydra from one that has been cut in half. But as this NPR video explains, there's a much more exciting possibility—the hydra might not die of old age. Take a look:
For more, you can read Mortality Patterns Suggest Lack of Senescence in Hydra (PDF), or check out other examples of biological immortality.