The problem I have is that everyone can rap the lean startup principles but when you ask who has successfully implemented lean startup, nobody says yes. How many people do you guys personally know that have successfully implemented lean startup?
It's a B2B product that has a couple substantial value adds over the competition. I created the MVP and got a core group of people in a paid beta. There were tons of bugs, but because of the unique value add I offered everyone was more than willing to put up for it and pay for it (and they knew they would get locked in at beta pricing for when it improved).
I was then able to quickly and easily iterate fixes and improvements, based on my vision as well as what I heard from my paying customers.
So on launch day I was already profitable, I knew what my users wanted, all the major bugs were completely sorted out, and I had ~100 paying beta users (all of whom were referred directly by me or by word of mouth through a beta user) who felt like they had a vested interest in the success of the product who helped me reach a critical mass.
With that being said I don't consider my company to be a lean startup (I hate labels like that), and I didn't set out to do a lean startup, but a lot of what I did is consistent with the lean startup approach.
But if you want a more traditional success story, I'd suggest Boxbee:
I was a mentor at a Lean Startup Machine event ~18 months ago where I met the founder. He had a very hazy idea, and over the course of the weekend refined it into something more focused. He then used the Lean Startup techniques to build a business that had revenue from the first month, and was quickly in the black. Relentlessly refining both the user experience and the back end, he found product-market fit and came up with a sustainable plan. And he did this all in his spare time, mainly with his own money.
A month or so ago he pitched it at Jason Calcanis's LAUNCH conference and took first place. Now he's raising money to build an infrastructure well beyond the initial MVP-grade product.
But you can also look at historical examples. It's important to note that none of the Lean Startup techniques are new. It's meant to be a crystallization of things that some people were doing before, but that had never really been talked about. At the local Lean Startup Circle event, we had the founders of Dropbox and Cafe Press talk about how they worked, and it was very much in the start-small-and-iterate-relentlessly style that the Lean Startup folks are promoting.
I don't mean to imply that there aren't any, but I just haven't heard of any.