As far as I am aware, female mammals cannot produce eggs (as in ova) after a certain age/maturation.
How could the female produce eggs for an unlimited amount of time, resulting in a 'queen' mammal similar to queen bees?
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The concept of a menopause is actually unusual, it's known to exist in the wild in 5 species, Humans, Orcas, Belugas, Narwhals, Short Finned Pilot Whales. While some other species exhibit menopause in captivity, others are definitely known not to e.g. cats and dogs.
In all other species the females are believed to remain fertile for their entire lives.
The only queen mammal I'm aware of is the naked mole rat.
The relationships between the queen and the breeding males may last for many years; other females are temporarily sterile. Queens live from 13 to 18 years, and are extremely hostile to other females behaving like queens, or producing hormones for becoming queens. When the queen dies, another female takes her place, sometimes after a violent struggle with her competitors. Once established, the new queen's body expands the space between the vertebrae in her backbone to become longer and ready to bear pups.
Women may make new eggs throughout their reproductive years—challenging a longstanding tenet that females are born with finite supplies, a new study says. The discovery may also lead to new avenues for improving women's health and fertility.
This means that the "finite number of eggs" reasoning behind menopause is not an absolute one in human biology, let alone non-human.
Women are born with approximately two million eggs in their ovaries, but about eleven thousand of them die every month prior to puberty.
Given one egg per month gives you a ballpark fertility age of 158,000 years (give or take 1,000 years).
If you're changing human biology enough that this is a normal lifespan, you can change it enough to keep the eggs fresh and only release one per month.
Basically the arrangement is that one or two females do all the breeding for a colony. The rest of the colony look after, feed the children, take care of the "queens", and so on. This is pretty much an identical set up to bees, but with fewer workers, and none of the haploid-ness.
There are a lot of intricacies to having a colony that is based around one or two females doing all the breeding that you can look into. For example avoiding inbreeding. However with regard to your specific concern:
Meerkats do not have infinite eggs, and don't need them. Infinite eggs are not required for animals that have finite lifespans. These animals only need to have more eggs than they can use in their lifetime.
The main issue here is that female mammals don't "produce" eggs, they mature them. every egg a female mammal will ever have is already present in the ovaries at birth.
To have a mammal with indefinite breeding age, and to reduce the negative effects of age on the egg (older eggs have had more time to become damaged), females must indeed produce eggs on the spot, similar to how the male produce sperm; sperm cells are always new and fresh, since they did not have the time to degrade the way eggs do.
The reason why women don't produce eggs for their entire life (making that National Geographic article linked in another answer somewhat dubious) is that this is not in any way in nature's interest. Read "nature" as either "evolution" or "fitness", however you like.
The incidence of several hereditary defects is directly linked to the mother's age, and goes up exponentially (not linearly). The reason for a 45 year old woman having such a high risk is that, well, her eggs and her DNA are 45 years old.
For that same reason, nature doesn't want 50 or 60 year olds to deliver, even if food supply and longeviety allowed for it. There's enough eggs in one woman for a couple of hundred thousand years, so 100 years wouldn't be a technical problem -- but nature doesn't want that. Because what they might deliver would have a high likelihood of being vastly inferior.
Nature isn't loving, kind, and altruistic -- she is a mean bitch. Nature doesn't cater for the inferior. If you aren't good, you are undesired (because you take away precious food and water from the more worthy).