Natalie “Nadya” Suleman, best known as “Octomom,” says Child Protective Services basically set up camp at her home in the early days after she brought home the octuplets … showing up at least once a week, yet never knocked her parenting.

Natalie — who’s going by her birth name now, instead of Nadya — came on “TMZ Live” to talk about what life is like for her and her 14 children now. Remember, she gave birth to the octuplets in 2009 after she was mother to 6 kids.

She opened up to us about all the public scrutiny she got after she first gave birth … including visits from CPS “at least once a week.”

All that being said, Natalie told us fame was the last thing she wanted when she had the octuplets, and she now says she wishes she had sued Dr. Michael Kamrava … the doctor who implanted 12 embryos during IVF.

He told her he only implanted 6, and eventually lost his medical license.

Suleman also gave us an inside look at her family life … saying her older children — the ones born before the octuplets — have an incredible work ethic. They’ve got jobs and are pursuing college — except for one of her kids who needs full-time care because of his autism diagnosis.

Nadya says she’s taking care of him full-time … though her tight-knit family pitches in when they can.

Suleman also pulled off another miracle in her family … ’cause her younger kids are all vegans — and, ya gotta hear why they decided to make the change.