I'll freeze anything, except raw eggs, and apparently there are even ways of freezing raw egg whites & egg yolks. I haven't tried freezing mayo yet, but it's only a matter of time. Oh, and freezing avocado doesn't work, but freezing guacamole does. Go figure.
I'll defrost and refreeze fish, raw or cooked. I even once froze a Subway sandwich, lettuce & tomato and all, to bring on a plane the next day. Did the texture suffer? Sure, but not as much as I expected, since lettuce tends to wilt and get soggy from mayo pretty quickly anyway.
But I also don't defrost things on the counter, or in the fridge. For me, things like veggies and meat go from the store, to cleaned & prepped & chopped (if appropriate), to frozen, to dumped into a pot or baking pan to be cooked. Then as soon as they're cool enough, any leftovers go back into the freezer.
It kind of depends on your tolerance for compromised textures. Defrosted cooked mushrooms can get rubbery. Defrosted potatoes can get mealy.
Pro tip: chop and roast potatoes, and then as soon as they're barely cool enough to touch, freeze 'em. Yeah, it strains the freezer a bit by putting hot food into it, but it preserves the potatoes' texture.
My advice is to experiment. See which foods suffer from defrosting in ways you don't like. See if there's a way around it -- if you chop mushrooms finely and make soup from them, the rubbery texture is obscured. Freeze a raw egg. Why not? See what happens. (Do it in a container, though, in case it cracks and leaks.)
I have yet to get sick from eating something that I froze even 3 times. And I rarely throw away food -- though I will admit that there are some questionable old containers in the back of my freezer that should probably be dumped, lol.
More tips?
Buy or grow fresh herbs, then clean, chop, dump into a container, and freeze. Then you can add a handful to things you're cooking.
Juice lemons, limes, oranges, and freeze the juice in ice cube trays (tiny cubes for the lemons & limes). Dump the cubes into a container. You can add them to your dishes, or defrost in a bowl if you want them liquid.
If you freeze roast potatoes or sweet potatoes, you can sometimes crisp up their outsides in the oven instead of the microwave.