Some really aggressive responses here... :/
tl;dr: Your birds are probably going to stop laying right around the same time they start developing painful and costly health issues. So you should probably cull if you don't want to run a costly and trying retirement home for hens.
This is how it goes down for me.
First, I try to avoid production breeds. They burn out faster. Heritage breeds lay nearly as well at 4 years old as they did at 2 years old. So there's not a solid reason for me to cull before they start hitting 5 years old or so...
When the really light egg layers slow down they don't exactly take a lot of feed to keep around anyhow.
The local predators have taken out far more chickens than I have ever had live to old age. Everything eats chickens.
Lastly, there's a BIG difference between natural lifespans for animals and captive lifespans. An example of this; rabbits live around 3-4 years in the wild and up to 12 years in captivity. Yet female rabbits between 6-9 years old have a 100% genetic chance of developing reproductive cancers and dying from it because there's NO genetic reason for them to have ANY inborn disease resistance or survivability beyond that point. A 5-year-old wild rabbit is unheard of. By design, rabbits live short lives and it's only our severe intervention that keeps them around longer.
Chickens are very similar. In captivity, their health starts to decline soon after they would have been eaten in the wild. That's somewhere around 4-6 years old, they start getting health problems more regularly, and their egg production falls significantly. They CAN live to be 10-12, but if they SHOULD is another question.
Dramatic egg decline isn't just because they're running out of eggs. It's also because they're not very healthy. When I find a bird that's only laying once a week they're almost guaranteed to pick up some sort of major problem soon after.
So I cull when I see the first signs of significant decline. A reduced egg rate but still getting 3 eggs a week isn't justification, like, that still covers the feed costs even. But when they start to drop below that I cull them. Every time I haven't done so I have regretted it. I have had bird keel over the next month or start developing parasites, or they suddenly move to the most picked on bird in the flock and are ostracized.... Something almost always goes wrong.
So ultimately, it's up to you if you want to cull your birds or not. But honestly, I would do so. Not only because it's good for your bottom line or because of soup or something... Which are all perfectly rational reasons for culling. But also because it's often the best thing you can do for their wellbeing as well.
If you're willing to resign yourself to costly vet bills you can probably keep them around to 10 years old. Some might even live that long in relatively good health. But if you're not, you're not going to have a lot of options on hand, and at least one of your birds is going to get sick or injured and force the issue. So it's best to make up your mind now.