Well, 3 is the new 2, you know. ;-)
Seriously though, constant stubborn whining can be exhausting for Mom. There are two basic "tricks" that you probably already know:
1. Be consistent. If whining works even ONCE, then any adorable kid knows that whining will work again eventually. They won't give up. You might not give in this time... but that little brain is thinking "Mwahh-hahh-ahha you WILL turn to the dark side and give it cookies."
2) They can't successfully whine at a Mommy they can't see. When my son's whining was grating on my last raw nerve, I lovingly-but-firmly placed him in an unpleasant time out spot, which was safely baby-gated out of my line-of-sight. (I could peek on him, but he didn't know it.) No toys, no tv, no people, no fun -- But he was allowed out the SECOND he was "happy again" and he knew it. A few hiccups later, the small voice would say "Out of time out? I happy now, Mommy."
Between consistency and yucky timeouts, my kid eventually gave up on the whining. Even for the most stubborn kid, if their action always (always!) brings them unpleasant results, they will stop doing that action.