My biggest failure was choosing the wrong spouse. When the crap hit the fan he couldn't take it. As I am unlikely to marry again and definitely not having more kids, I haven't learned anything useful from that.
My biggest failure that I learned a huge amount from was a business I started in my 20's. I'd already been successful at running my own businesses and selling them for a profit, so I had a good idea, some good people, and some investors. We had a great time for a while but it turned out that we wouldn't be able to manufacture our product at a price we could sell the things for, and it went belly up.
But I learned so much at that. How the biz funding game worked. How to look at investment money as a tool, not as money - and that those investors didn't really care that we lost everything, they expected most of their high-risk investments to go belly-up. I made a lot of good friends & contacts, learned a huge amount about my field, and found out what's important to me - what to focus on and what to let go of.
Mainly I gained credibility. Amazing what failure can do for your reputation sometimes.