The vast majority of animals undergo degenerative aging: it is evident, easy to measure, and well recorded. The remaining tiny minority may undergo degenerative aging, but due to a combination of insufficient study and individual robustness throughout life researchers can't yet accurately determine the outer limits to their life spans, or how fast they age, or definitively how and why their aging is different from ours.
Lobsters fall into this category (though possibly not for very much longer), as do hydra, some clams and mussels, and a handful of others. It's probably worth remarking on the fact that if naked mole rats were much less studied, they might also be on the list - but since researchers have colonies of thousands of naked mole rats in laboratories and are intensely focused on their peculiar biology we have a much more comprehensive idea as to how they age, what their maximum life span is, and so on. Conversely if all those naked mole rats were instead arctic quahog clams, then I'd wager clams wouldn't be on the list above. To a certain degree the list of potentially ageless animals is as much a list of ignorance as it is of interesting and resilient species.
The vast majority of animals age because aging is a necessary evolutionary strategy for success in a changing world, or at least that is the common view:
When conditions change, a senescent species can drive immortal competitors to extinction. This counter-intuitive result arises from the pruning caused by the death of elder individuals. When there is change and mutation, each generation is slightly better adapted to the new conditions, but some older individuals survive by random chance. Senescence can eliminate those from the genetic pool.
From the perspective of individuals this is a race to the bottom, wherein lineages that result in more harm to their own members prosper and spread at the expense of those that do not. Successful it may be, and no doubt the bone mountain of history is a necessary precondition for you and I to be standing here today, but that doesn't mean it is a good thing here and now. In particular, in the decades ahead you and I will suffer decades of pain and privation leading up to an ignominious, undignified death assuming nothing is done about the situation. This is an age of biotechnology, however: now that we can do something about it, we should be doing our utmost to get rid of degenerative aging, and the suffering and death it causes, and take the reins of the future into our own hands.
Today I noticed a slightly different way of looking at the near-universality of aging as an outcome of evolutionary selection. It comes from an author who is very much in the programmed aging camp when it comes to how and why aging progresses in an individual, but for the purposes of this argument about the evolution of aging that doesn't matter all that much:
If it weren't for aging, the only way that individuals would die would be by starvation, by diseases, and by predation. All three of these tend to be "clumpy" - that is to say that either no one is dying or everyone is dying at once. Until food species are exhausted, there is no starvation; but then there is a famine, and everyone dies at once. If a disease strikes a community in which everyone is at the peak of their immunological fitness, then either everyone can fend it off, or else everyone dies in an epidemic. And without aging, even death by predation would be very clumpy. Many large predators are just fast enough to catch the aging, crippled prey individuals. If this were not so, then either all the prey would be vulnerable to predators, or none of them would be. There could be no lasting balance between predators and prey.
Without aging, it is difficult for nature to put together a stable ecosystem. Populations are either rising exponentially or collapsing to zero. With aging, it becomes possible to balance birth and death rates, and population growth and subsequent crashes are tamed sufficiently that ecosystems may persist. This is the evolutionary meaning of aging: Aging is a group-selected adaptation for the purpose of damping the wild swings in death rate to which natural populations are prone. Aging helps to make possible stable ecosystems.
Aging as a damping function introduced into population dynamics is an interesting way of looking at it. Starting from that purely mechanistic perspective it makes one wonder why aging came to near-universally dominate as the source of that damping function - why are there no other viable alternatives to achieve that result in the natural world?
From the perspective of aging as a process caused by an accumulation of cellular and molecular damage, rather than a process caused by evolved genetic programming, longevity evolves when evolutionary success is predicated by living longer, so that better repair and maintenance mechanisms are favored. In this view, aging was a given from day one of the first evolving organism, as all complex systems suffer damage in the course of operation. Even bacteria age, putting the origins of aging far, far back in the deep past: when the first multicellular organisms emerged the fact of degenerative aging was already baked into the mix. The only question from there forward was whether and how evolutionary circumstances would lead to biological mechanisms that mitigated or repaired the damage that causes aging.