Interestingly, you can have unboundedly many children with only quadratic population growth, so long as they are exponentially spaced. For example, give each newborn sentient a resource token, which can be used after the age of maturity (say, 100 years or so) to fund a child. Additionally, in the years 2^i every living sentient is given an extra resource token. One can show there is at most quadratic growth in the number of resource tokens. By adjusting the exponent in 2^i we can get growth O(n^{1+p}) for any nonnegative real p.
Interestingly, you can have unboundedly many children with only quadratic population growth, so long as they are exponentially spaced. For example, give each newborn sentient a resource token, which can be used after the age of maturity (say, 100 years or so) to fund a child. Additionally, in the years 2^i every living sentient is given an extra resource token. One can show there is at most quadratic growth in the number of resource tokens. By adjusting the exponent in 2^i we can get growth O(n^{1+p}) for any nonnegative real p.