DNA (Photo credit: Josh*m)
A very good question, and one that will be added to the “Great Mysteries of Nature Nobody Seems to be Thinking About” post.
The oldest living things we know of are the gnarly bristlecones that have a couple three millennia behind them, but even they succumb.
Why? Why not just go on living forever?
The answer may be the nature of the light that sustains us. The same photons that provide the energy for photosynthesis and presumably the random aggregations of chemicals that engendered life in the first place also break down complex chemicals and damage DNA.
Chaos and entropy take their toll as well. Living things can be thought of as well repaired machines. In the sense that living things are mere vehicles for DNA to reproduce itself, the vehicles become old cars whose cumulative repair costs exceed the cost of producing new ones.
Within this framework, organisms have developed very different strategies, from small, rapidly reproducing quickly colonizing “weeds” with low energy and organizational needs and short life spans; to large, slower reproducing slower colonizing “sequoias” with large energy and organizational requirements that live a very long time.
My old truck has 650,000 miles on it and it refused to start last night. The symptoms were very strange and got me worrying about nasty electrical repairs. It even got me thinking of buying a new one. Turned out to be only corrosion at the battery connection. Is that chaos or entropy?
geosciencebigpicture 
Oh, I’ve been thinking about this question very deeply. When I discovered that I am Immortal, I realized that was not a subject that could be discussed generally. Nor is it a subject that is, well, lets just say, believable.
“Immortal” is one of those words that’s bandied about without much thought. People can use it to mean that their body will live forever, or their soul will. But there are all sorts of nagging questions about this. Does immortality mean you can’t be killed, or just that your body will not age & decay if left to natural processes? If you ran out of air, would you live in a state of anoxic agony forever? Or could you go on without air as normal?
So I’m curious …. what do you mean by immortal?
In a way I also felt immortal when I was younger. Not so much rationally as just the exuberance of boundless energy, as if anything were still possible and death like some bill payable, could be postponed a long while and negotiated down to something reasonable when the time came.