Hi!
Try it and see - when you've got a planet of 2.5 billion producing 500 million pop growth per turn (which may be possible if the fertility clinic bonus is applied after the cap), then let us know.
I did. With 9 Fertility Clinics I grew there 594M pop. Other findings regarding FCs:
1) Ferility clinic seems to increase growth in the turn it is built.
2) each clinic increases base growth (3% or 75M) by 25%, with additive effect. The only bonus affecting it's efficiency is approval on a planet. No other bonuses apply.
Conclusion: basicaly a FC adds each turn 38M pop at 100% approval, and 19M below 75 approval. IMO it's not worth to build it.
Here are raw data. I obtained them with race growth bonus 72% by helding a homeworld at 100% approval, and letting the planet build several Fertility Clinics. First column notes the turn in which the FC was built, second is population, third is the difference of population to previous turn (growth), and the last is the change of growth to the previous turn.
new FC pop growth dif-FC
------+-------+-------+-------
4258
4516 258
f 4810 294 36
5104 294 0
f 5436 332 38
f 5806 370 38
6176 370 0
6582 406 36
f 6988 406 0
7432 444 38
f 7876 444 0
8358 482 38
f 8840 482 0
9360 520 38
9882 522 2
f 10400 518 -4
10958 558 40
11516 558 0
The differences in last column are probably because of a float pop number rounding to display as an integer.
BR, Iztok