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These are the three (and a half) generations of the Rich family before Filthy himself.

Starting from the far right with the oldest generation and going down the line. For slightly further context https://mlp.fandom.com/wiki/Rich_family#Extended_family

Strikin' Rich and her husband Seed Investor
A long, long time ago, before Ponyville was even a twinkle in old timey ponies’ eyes, a couple who saw the stars in their future was hard at work. Strikin' was the charm of the operation who could talk a crystal pony into buying snow. Growing up in an earth pony family gave her a lot of knowledge the half-pegasus Seed Investor didn’t have. The pony kinds were united but some still turned a prejudiced eye to mixed ponies, leaving him in the dark despite his name. He crash landed on her farm one day, and as she helped patch him up, they talked all about their interests, leaving Seed to become the brains and the labor of the operation. The Rich family happily accepted the stallion that would later become their son-in-law and the two expanded their farm as much as they could, planting lots of different fruits and vegetables. One day, still working into the late days of her pregnancy, Strikin’ gave birth to a colt in the onion patch who they named Stinkin’ Rich.

Stinkin' Rich and his wife Fertile Fields

Stinkin’ Rich grew up deeply loved, the best of both his parents’ traits and a dedicated farm boy. Although, the labor of the farm wasn’t really what he wanted. His family farm was quite successful and he never had to go hungry, but they had hit an impasse. Ma and Pa were too old to truly get any further with the life they had built for themselves so Stinkin’ had an idea. Leaving his parents teary-eyed but proud, he made his way across Equestria, meeting some ponies not too unlike his own family selling amazing rainbow jam. It was the best he had ever tasted and he saw a huge opportunity. He would supply the Apples with the last harvest he had brought with him and whatever extras Stinkin’ brought back when he visited his folks, in addition to selling whatever they (and the other pony artisans and farmers) would make. But he chose to settle full-time there and earn his own place in the town that would come to be known as Ponyville. As the town grew, so did his wealth, yet he made sure to stay right at the register. Stink valued the community and wanted to stay the kind, helpful face. A certain demure mare coming into the shop every Friday was definitely a perk. She didn’t talk much, and every time their hooves accidentally touched while exchanging bits she’d run away giggling, so he set out to come to her first. Everypony knew everypony back in those days, so he was able to find out where she lived and showed up at her doorstep with flowers in his teeth and his hat against his chest. From her father calling her to the door, he learned her name was Fertile Fields, and he asked as sweet as honey if she could do him the honor of allowing him to get to know her. He fell even more in love with her as he learned the magic Field could make in the kitchen. The pair were practically inseparable and found no better company than in each other. Which is why they decided to build a cabin in the nearby Whitetail Woods where they could live and grow their family away from the hustle and bustle of the Rich family store.

Impossibly Rich and her husband Steel Baron with their son Incredibly Rich
Impossibly grew up at the cabin her parents built learning recipes from her Ma and helping her Pa with repairs. She was definitely spoiled and didn’t fully understand the hard work that got her family where they were but Possie wasn’t a brat, just an excitable curious filly with a wild imagination. She would make up stories to entertain her family and the patrons at the store, an adorable delight to everypony she met. At his point Richs Barnyard Bargains made her family well…impossibly rich ponies, drawing the eye of other wealthy families. Her father was approached by Iron Strong, a stallion whose family made their money by building the railway, a stallion with a colt her age. Iron felt that their families could be stronger together. Stinkin and Fields were reluctant but were open to the idea, refusing to refer to the pair as betrothed unless they naturally came together. So Impossibly found herself with a new friend, when he wasn’t busy with his own studies the two would frolick and play in the woods around the cabin. It was a match made in heaven. Not just because the pair got along but because Impossibly Rich never found herself having an acumen for business but Steel did, meaning the family business would be inherited into good hooves and Stinkin’ Richs daughter would be happy. The pair knew each other inside and out, being friends since foalhood after all, so they were considered quite the power couple and just as inseparable as Possies own parents. They provided their son with everything he could ever want but neglected to pass on the importance of love and community that had been sustaining the family for generations. No matter what they or their army of nannies and butlers tried, Incredibly Rich put money and power over everything else. He never planned on getting married himself until he was practically forced to.

The family saw his naming of his son, Filthy Rich, as an homage to the stallion who made their comfortable lifestyle possible but Incredibly knew inside it was his only way of making his feelings clear. Growing up he made no effort to interact with his son outside of teaching him the way a business should run. This is how we got to the stallion we know now who often makes choices that seem as though he cares for his business above everything else, above family and nature. He too married a mare he truly loved but who he thought would be best for his business, give him a kick in the flank when he was being too soft. After these literal centuries of history Filthy Rich is a truly a good hearted stallion who had a multi-million dollar business placed in his lap the day he was born and now has to work on fixing his own mistakes as a father.