Well it depends on the situation or on the person
Actually i onced asked a rich guy (my classmate in college) before if is it fun being wealthy.
He said at first it is but as he grew older, he's getting tired of it and he wanted to try something born rich people didn't experienced (being poor ) so he hang out with me (yes, i'm a poor guy) and showed him our(poor people) world. He said it was hard but it was fun and he really admire me on how i can survive on that situation
don't think he was really born rich then.
People who are truly born rich really have no idea they are rich until they're an adult. Even then they might not realize it. And I'm not talking about their parents hiding their wealth or something, it's just that children takes their own situations as natural, thinking that's the way the world is. Once a rich man took his daughter on a commercial first class flight, and the little girl was really surprised, asking her daddy "daddy, why are you bringing hundreds of friends with us?" Since she's only ever been on private jets, she thought that all the passengers were her dad's friends.
So they won't think if it's fun or not fun, they'll think it's natural and the way things are/should be.
Which is why sometimes i wish i was born wealthy and had the easy life all my life, but if I am able to succeed and become wealthy myself, I'll probably be able to enjoy it a lot more because I have something to contrast it with.