There are many supposed paths to immortality, though few have been tested, and those that have been tried have just as often proved to be mere flights of fancy, fashionable cure-alls with no real effect besides the status they grant those able to acquire them.
It is rumored that several pools and springs throughout Tamriel can grant immortality. These places are often hidden in the most inaccessible and hard to reach places. Such places are rumored to exist deep in the swamps of Shadowfen, on the highest peaks of Rivenspire, and in an ancient glade of Malabal Tor. Explorers have sought out these places for centuries, and while many have returned with stories of their success, not one has yet thought to mark the locations on a map.
Among nobles of the First Era, drinking the honey of the Isgareth Bee of Auridon was said to grant limited immortality, though one needed to continue to eat the honey in order to maintain the effect. Some say this led directly to Isgareth Bee's extinction, as indiscriminate men destroyed whole hives in order to more quickly harvest the precious golden substance and sell it at a high price to the foolish and fashionable. Though it is also said that Altmeri kings and queens maintain a private hive.
Another means of achieving immortality is said to exist, but it has only been attempted by the most volatile and unstable. It's also forbidden, both by governments across Tamriel and by the Mages Guild. It's said to involve the binding of a Daedra through blood sacrifice.
For so long as the Daedra can be held captive, life may be prolonged. It is unknown whether any who have attempted this method have been able to see it through to its end. It's more likely that the Daedra breaks loose and kills the attempter, or that the attempter destroys himself through his own madness.