While in testes true "stem cells" (i.e., the spermatogonia) remain present and actively divide to maintain the supply of spermatozoa indefinitely, the situation in the ovary is very different. The stem cells of eggs, the oogonia, are not present in mammals at the time of birth.
In the course of embryonic development, all of the oogonia that are ever going to develop from the germ cell line do so, and then, as the ovary forms, further differentiate into the next stage, oocytes. There are two stages of oocyte development, and in primordial follicles, you will find not oogonia, but primary oocytes. These are arrested in development at the stage of first meiotic prophase, during life in utero. By the time of birth they have all reached this state of suspended animation, which lasts for years (or even decades in long lived animals).
Female mammals are therefore born with all the follicles they're ever going to have, and each one contains a primary oocyte. There are substantial numbers of these. In humans, for example, the total number of primordial follicles present at birth can easily reach 500,000 or so. These follicles and their oocytes remain "asleep" until the onset of puberty, then the hormonal changes that occur cause them to "waken" from their dormancy and begin to develop.
But in fact most of them never develop at all. Again to take humans as an example: women have a much longer reproductive life than most other mammals, but even so, if a woman ovulates once a month from the ages of 12 to 50, she will produce only about 400-500 matured gametes potentially capable of fertilization.
Among domestic animals, in non-litter-producing species with considerably shorter reproductive lives (e.g., cattle and horses) the number of naturally occurring ovulations is correspondingly smaller, though there may be as many follicles available to start with. Even in species which routinely produce litters, the "wastage" of follicles is very high. If a bitch has as many as ten litters of 10 pups each (a highly unusual number for most breeds) she's only going to use four-one thousandths of a percent of her follicles. Thus, in all species only a minuscule fraction of those follicles will ever get past the primordial stage.
This vast excess of potentially fertilizable eggs is the basis for the technique of superovulation. By appropriate hormonal manipulation, many more follicles can be caused to develop and release eggs than would otherwise be the case. These can then be collected, inseminated in vitro, and distributed into surrogate "mothers" for development.
The trick works better in some species than in others. It's widely practiced in cattle and in humans. Thanks to the miracles of modern science, it's not only possible, it's routine for a cow to bear calves to which she has no genetic relationship. In some instances she may even bear a "calf" of a different species! The methodology has been used to produce calves of the gaur (an endangered wild form of cattle from the Indian subcontinent) with Holstein "mothers."
Because superovulation and embryo transfer works very well in humans, some odd situations have come about. There are many documented instances of surrogate mothers "renting" their uteri to hold a child for women who could not do it themselves; and in what is perhaps the most bizarre twist of this tale, one woman in the Republic of South Africa has provided a surrogate womb for her daughter's child! Thus the biological mother and the birth mother are different, and the birth mother is really the child's biological grandmother. It's quite clear from these examples that superovulation and in vitro fertilization offer chances for immense profit to cattle ranchers, fertility clinics, the makers of liquid nitrogen, and attorneys who are creative thinkers. The situation also raises some significant questions of medical ethics, most of which have yet to be answered.