Well, this needs to be broken down into two parts. Do Crocodilians age (undergo senescence), and are Crocodilians immortal (will only die of external causes)?
Are Crocodilians immortal?
- No. They appear to live about as long as humans before they die.
Measuring crocodile age is unreliable, although several techniques are used to derive a reasonable guess. The most common method is to measure lamellar growth rings in bones and teeth—each ring corresponds to a change in growth rate which typically occurs once a year between dry and wet seasons. Bearing these inaccuracies in mind, the oldest crocodilians appear to be the largest species. C. porosus is estimated to live around 70 years on average, with limited evidence of some individuals exceeding 100 years. One of the oldest crocodiles recorded died in a zoo in Russia. A male freshwater crocodile at the Australia Zoo is estimated to be 130 years old. He was rescued from the wild by Bob Irwin and Steve Irwin after being shot twice by hunters. As a result of the shootings, this crocodile (known affectionately as "Mr. Freshy") has lost his right eye.
Do Crocodilians undergo senescence (show signs of aging)? Well, if this study from "Gerontology" written by Patnaik BK in 1994 is to believed... Maybe not.
Evidences and mechanisms of rapid or negligible senescence in reptiles are still fragmentary and unclear... neither the increase in mortality rate and accumulation of lipofuscin nor the reproductive senility have been shown conclusively in ageing reptile populations.
So, while Crocodiles and Alligators (both Crocodilians) definitely have a finite lifespan, because they just continue to grow it's hard to tell how long they have left until the day they die.