Not that sanity is my specialty, but the only sane and non-destructive reasons I can think of for this is to either recover from a massive population loss (the Gerbil Apocalypse of 2134 wipes out anywhere between 21% to 93% of Earth's humans in 98% of all sampled parallel timelines) or to colonize a new world.
Currently in this timeline, the nation of Niger has a fertility rate just under 7. This is the current record and is about where the USA was in 1800. Let's use that as our base point and see where we can push it. For a simple start, I'm mostly focusing on plain vanilla traditional reproductive methods for plain vanilla couples, but other options will also be considered.
The original question was for people with plenty of money, but as many have pointed out, wealth alone is a very effective fertility suppressant. Instead, our optimal population explosion society needs to be post scarcity. Good tasting healthy food leads to better pregnancy outcomes. Having to work for food is a distraction. Good shelter also needs to be readily and easily available to keep parents and children safe and warm. So, we're going to need automated food production as well as either automated housing construction or super easy DIY house and expansion room kits.
This alone won't make the average happy couple have 7 children. We need to add servants. Since we're in a post scarcity attempt at utopia, robotic servants are the logical choice. Forms may vary, but certain functions are critical. We need doc bots to make sure all goes as well as possible with pregnancies. We need nanny bots to take care of the upcoming tidal wave of children. We need house cleaning bots to pick up after all these kids. We'll need teacher bots to tame, I mean educate all these children. We'll need psychiatrist bots (probably doubling as bartender bots) to help the parents to not completely lose it too.
All of the above makes having a lot of children less of a burden, but we still need more motivation. Our parents must be educated (brainwashed) to understand that increasing the population is an important duty for all members of this society. Their children need to also be taught this attitude. The success of this psychological conditioning is critical. Even a post scarcity society has social status and other rewards. Those who produce more children are rewarded. Those who fail (at least by choice) will be stigmatized.
I think this would be enough to get fertility to 7.
To go higher:
Aim for multiple births. Fertility drugs is one choice. Implanting 3 or 4 embryos for every IVF attempt should get increase the odds too.
With some surgery a hormone therapy, it may be possible to emplant embryos in males. Large scale studies would be needed to look for possible side effects in children gestated this way. Possible effects on a male body from one or many pregnancies are currently unknown.
Nursing babies may be a wonderful thing, but does temporarily reduce the mothers chances of conceiving another child. If nursing is permitted, limit it to 3 months at most.
Healthy women in developed countries remain fertile longer than those in the past or in less developed countries. Anything that can be done to add a few more fertile years adds more potential for racking up the baby count.
Human choice would still be involved to some degree, but I belive under these conditions, average fertility could easily exceed 10 babies per couple.
If this population boom is a planned event and all other priorities are rescinded, shift the male/female ratio (can be pushed a little by sperm separation techniques, can be pushed as far as you want via selecting which embryos to implant for IVF). If we only have an average of 4 babies per woman, we can jump from 10 adults producing 35 total babies (7 per couple) to 63 total babies by replacing 4 of the 5 males with females. If we can completely dump males, the total would be 70 babies.
But there is one more option tha ends all limits. With a reasonable starting number of embryos (or eggs and sperm), some sort of safe and effective artificial wombs would free women in a situation like this from the unenviable role of baby factories. In this scenario, the potential limiting factors would be the number of artificial wombs, the production of factories for food, housing, and servant robots, and just how many dozen (hundred? thousand?!?) children running around per adult human would be acceptable.