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Most people give about a thumbnail size piece, I've never had a problem with them eating too much but it's always best to begin with tiny pieces, also if they stash a small piece it will just dry out rather than go mouldy like a larger piece of veg might.
Any kind of potato needs to be cooked, I give green beans cooked although I think they can have them raw but nobody here likes them that way!
Other than that it's generally best to give raw although the occasional piece of cooked veg is fine.