I can't talk first hand on what being financially rich feels like but if you'd still like my two cents, here they are:
Depression is not necessarily something that appeals to logic. I remember that part of my depressed state was constantly beating myself up because compared to other people I was in a very privileged position. I had friends and family that cared about me. I could understand that, but wasn't something that gave me comfort, it was something that made me feel WORSE. It was a self loathing cycle. The more I realised the more I hated myself for feeling the way I did "for no reason".
Now back to having all the money you could need, without taking in account that there is still sickness that are deadly and have no cure as potential candidates, I think the depression could come because of similar insecurities. While money gives you a way to alleviate a lot of the pressures to get by (don't have to worry for bills and stuff) and you can get the best professional help you ever wanted, have balanced diet and stuff... We're still social creatures and we value connection.
Many of us can cherry pick how 'easy' the beautiful and/rich people have it, yet we also hear how that can make said people find it hard to trust other people to be as close to them and treating them the way they do for themselves and not just because they want a slice of the pie. So, I am assuming that a rich person that doesn't have people that they can truly be vulnerable with can really feel isolated. Not to mention all the bad experiences of opening with someone just to find out they were trying to use them. I am sure that several similar experiences would wear a person down.
It might even be worse if you become rich later in life, because the shock of how everybody treats you when they learn you are rich can give you whiplash; and the taxes and the places/clothes/food etc you didn't have access to before, but here you are now affording a single frivolous meal/piece of cloth/accessory/etc that costs so much more than what you could do in a normal 9-5 job, and that is probably just one in many things you will get can really mess with your head.