Netflix is the first example that springs to mind. They're not purely a web app, but they're the only thing I'm paying for besides 37signals.
I might pay one of these domain-name-brainstorming sites that are popping up, not sure yet. Buxfer.com also looks intriguing, not sure if they charge yet.
'... Almost everything on WordPress.com is free, and things that are currently free will remain free in the future, but we do offer paid a la carte upgrades for things like CSS editing and custom domains. How do we pay for everything? WordPress.com is run by Automattic which currently makes money from the aforementioned upgrades, blog services, Akismet anti-spam technology, and hosting partnerships. ...' ~ http://wordpress.com/advanced-services and http://wordpress.com/about
stands corrected. Looks like there is more than what first appeared. I notice they don't advertise this much.
2 - Premium news sites. You see a lot of these with financial sites such as theflyonthewall.com, briefing.com etc.
3 - Quality tutorial sites. lynda.com
4 - I started paying up for an ad free experience and premium features for email.before gmail took off.
5 - Job listings on craigslist.
6 - Membership sites. Too many to list.