There are animals that don't "age"[1] so of course it's programmed. There's a balance between how often you want to recombine genetics and how long you want to be able to remember. It seems like a better system would have humans decline differently (i.e. no dementia) but having people grow old is almost certainly beneficial to the species.
[1] Their risk of death does not increase after reaching adulthood. I.e. every year after adulthood has the same risk of unexpected death. Humans and other animals that age have their risks of death climb in "old" age.
That you can go 300km/h in an F1 car, doesn't mean you can go that fast on a bicycle, even though they're both in the genus "wheel-based vehicles for human transport". The jellyfish is a rather different animal. Trees can grow very old, but there's a reason why you didn't use those as an example: too far from human biology. That jellyfish may be at a similar distance.
It would be a scientific process to find the limits, but I don't see a scientific reason to find out how to overcome them. Science' job description is oberving and explaining, not altering nature.
We have the ability to convert to becoming immortal. It actually happens to some of us. It sucks though. We call this cancer. Turns out the immortal cells become selfish. But you can scrape some off yourself, throw them in a petri dish, supplement growth media, and they will divide forever.
IANAB but lobsters are worth a look. They apparently do not get slower or weaker or even lose fertility as they age. And apparently Greenland Sharks might live for 300-500 years, are immune to cancers, and their DNA can repair itself. Maybe leaving the ocean was our ancestors' mistake.
Greenland sharks are interesting because they do everything slowly, including aging. Our best guess is that pregnant Greenland sharks have at least a decade-long gestation period. Which seems like an adaptation to cold and relatively barren environments?
Naked mole rats are social animals that live in colonies. There are differences of course, but I do not think the evidence suggests that aging is a 'requirement' for complexity. The simplest explanation is that aging is an evolutionary adaption that gives species that age an advantage. Human cells obviously possess the potential to persist past their programmed deaths - that's what cancer is. So I think it's obvious there are no biological limits to us stopping our physical age at whatever stage we want - but I'm equally sure that would involve changes to the experience of being human we may not like (perhaps being in the body of a 20 year old means being in the mind of a 20 year old, for instance).
They seem to have a biological clock and are just more isolated from the effects of aging. Humans are similar that we live three to four times as long as other great apes.
The best example of human cells having programmed death is simply the main exception: offspring.
Children are made from a cell of the mother, combined with some DNA of the father. It is not created out of nothing or from scratch, it quite literally just one of the cells of the mother. So if a child survives one of the cells of the mother survives the mother's death.
Obviously the process involves resetting the "death counter". In fact there's a disease where that step seems to have gotten skipped: progeria.
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[1] Their risk of death does not increase after reaching adulthood. I.e. every year after adulthood has the same risk of unexpected death. Humans and other animals that age have their risks of death climb in "old" age.
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With medicine it's really the entire point, it's science we learn to improve humanity.
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Children are made from a cell of the mother, combined with some DNA of the father. It is not created out of nothing or from scratch, it quite literally just one of the cells of the mother. So if a child survives one of the cells of the mother survives the mother's death.
Obviously the process involves resetting the "death counter". In fact there's a disease where that step seems to have gotten skipped: progeria.