The individual members of a very small number of species are functionally immortal. These are all lower animals that exhibit a profound capacity for regeneration and lack sophisticated nervous systems, such as hydra or jellyfish. A hydra is essentially a hunger-motivated bundle of stem cells, at least from the perspective of the mammalian world of limited and regimented tissue regeneration. Absent predation or accident these animals do not exhibit any increase in mortality rate over time. Proving that to be the case is actually quite challenging. For hydra, for example, researchers conducted a long-running experiment to assess mortality that involved hundreds of carefully tended animals kept for years.
We mortal individuals are offshoots of an immortal germline cell lineage; it isn't that much of a stretch to envisage one of those offshoots, such as hydra, extending the germline concept to a mass that consists of a few tens of thousands of cells. That transition happens during embryonic development for higher species, after all. Loss of immortality appears to arise once a species starts in on larger cell counts in the body, or a complex nervous system that stores state. Something about those characteristics is largely incompatible with an exceptional regenerative capacity, a body that is largely stem cells.
Returning to the theme of it being a time-consuming exercise to establish whether or not a species has a lifespan, let alone putting definitive numbers to that lifespan, we might look at how little is known of lobsters. Like many marine species, far less is known about lobster aging and lobster lifespan than most people might guess, given the size of the industries focused on farming lobsters. Until very recently, it wasn't even possible to measure the age of a lobster reliably. This is one of many species that exhibit negligible senescence: the appearance of few to none of the usual evident signs of aging over their life span. There is a rapid decline at the end, but up until that point there is little sign of that impending fate. Individuals remain vigorous and capable of reproduction all the way through. None of these species are expected to be actually ageless, given what is known of the biochemistry and physiology of higher animals (they are not roving bundles of stem cells, after all), but proving that hypothesis that becomes ever harder as species life span increases. After a certain point it becomes impractical to sit around and wait, and lobster life span, what we know of it, is well past that line in the sand.
Most research into animal aging proceeds at the sedate pace of a poorly funded field of study. This slowly progressing community is one in which scientists intermittently establish that, yes, yet another higher species - usually a marine species, as that is where the biggest gaps in knowledge are to be found - appears to be negligibly senescent. Today's example is the bigmouth buffalo, a well known freshwater fish species. Beyond reinforcing the point that evolution produces strange and interesting outcomes, what could result from this field? It tells us that in the very long term, re-engineering humans to have a better cellular metabolism that is less prone to degenerative aging is a viable project. If an outcome exists, that outcome can in principle be replicated. But in the near term, meaning the next few decades in this context, it is very unclear that understanding any of this biochemistry and its interactions with aging will (or could) lead to medical technology that will help unmodified humans resist or reverse aging.
Although the pace of senescence varies considerably, the physiological systems that contribute to different patterns of senescence are not well understood, especially in long-lived vertebrates. Long-lived bony fish (i.e., Class Osteichthyes) are a particularly useful model for studies of senescence because they can readily be aged and exhibit some of the longest lifespans among vertebrates. In this study we examined the potential relationship between age and multiple physiological systems including: stress levels, immune function, and telomere length in individuals ranging in age from 2 to 99 years old in bigmouth buffalo (Ictiobus cyprinellus), the oldest known freshwater teleost fish.
Contrary to expectation, we did not find any evidence for age-related declines in these physiological systems. Instead, older fish appeared to be less stressed and had greater immunity than younger fish, suggesting age-related improvements rather than declines in these systems. There was no significant effect of age on telomeres, but individuals that may be more stressed had shorter telomeres. Taken together, these findings suggest that bigmouth buffalo exhibit negligible senescence in multiple physiological systems despite living for nearly a century.