I make Nomad List (http://nomadlist.com). Not sure if it's succesful, that's up to you to decide. But it has revenue, profit, 3,000+ paying members, premium advertisers, was #1 on here and Product Hunt, been on Reddit's frontpage and has a decent reach in the remote work community.
[3] I don't know if these two businesses can be considered startups. To me startup is something defined. Not very specifically, but not very loosely either. The feeling I'm getting reading HN is that any (self-perceived) modern or pseudo-modern tech-company that doesn't have reason to market itself as a structured corporation poses as a startup.
There is a really good book out there on one person startups called the $100 startup by Chris Guillebeau - it gives loads of examples of startups around the world, which were started by one person (http://amzn.to/1Kb63VH)
I have read that book; Its definition of a startup was affiliate marketing or selling courses on selling. There was few examples of real businesses like Coffeeshops or a Photographer but I don't think we consider those startups.
When you say one person, do you mean didn't have a co-founder/team when they started? Or still only one person to this day? If the former, then Balsamiq was started by one guy.
One person, no funding.