It isn't known. But probably not for many decades.
However there are other factors that prevent them from growing exponentially and evidence for all of it.
Most Babies don't survive - While the Omicronians are upset, they don't seem nearly as upset as we would be under similar circumstances. This is because, like creatures on Earth that leave large clutches of babies abandoned, only a small fraction will survive to adulthood. On this planet alone there are probably all of these threats and more:
- Vicious Animals
- Bad Weather
- Natural Disasters like floods or fires
- Alien races discovering you are delicious
They commit matricide, and probably patricide - "When our species grows up, we eat our moms." The Omicronians eventually eat their parents when they mature. Exactly when isn't clear, but it is implied that it is connected with gaining maturity. Call it an Omicronian Bar Mitzvah. This also implies a near steady-state population for the Omicronians, which may be why they prefer to conquer and then leave planets. And may also be the reason why they have not taken over large parts of the universe.
Larger issues with Cannibalism - If the Omicronians eat their parents, they possibly also eat others of their own race. (Hence the cannibalism bullet on the above list.) And that too would serve to reduce their population and standing army.
Long maturation time - As I mention above. The Omicronians certainly age slowly. Otherwise Lrrr and Ndnd would have been consumed many years ago by their children.
Update: In Light of "T.: The Terrestrial" - While we know now that Omicronians age similarly to humans at least from awakened poppler stage until teenagerdom, we still don't really know how long the poppler stage is and we don't know how long it is until they reach the mother-eating stage.