Yams, potatoes, rice and avocados come to mind. Perhaps make a green smoothie that's palatable, and include a lot of greens in that.
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I was also thinking about making a smoothie because that might just work for your child. Also if they are someone that is picky, then I would suggest that you maybe not tell them what it is because sometimes they don't even know that they hate what they're eating.
After raising twin boys (now 22yo) and now parenting another 2-1/2 yo, I would say make sure you all eat the same healthy foods and don't offer substitutions at meals if they're picky, as that just builds a bad habit for them. They won't starve and may start with a tantrum or two for a few days, but once they realize it isn't getting anywhere, they stop complaining and eat what you give them.
Skeletor is also right. Plenty of fiber in non-greens also, like sweet potatoes and avocados.
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I just started giving my kids Fiber Gummies. The only ones my 3 and 5 yr olds will take. The brand is Fiber Advance Gummies with the fruit flavor, package shows raspberries, strawberrys and blueberries on it. They are very picky. You can get it on Amazon or at Walmart.
Chia seed pudding? I take about 24 oz of coconut milk, several tbsp. of chia seeds, several drops of liquid stevia, cinnamon, vanilla, mix it up and put it in the fridge overnight. It makes a great dessert that's sweet without all the sugar. They'd get fiber and a sweet dessert. I bet your kids wouldn't even know it's good for them. There should be roughly 5 g of fiber in each tbsp.
You could also add Jarrow Inulin/FOS to just about any beverage you give them. It's a great prebiotic and dissolves easily leaving next to no taste.