Despite being generally suspicious of any branding relating to "the cloud", I actually do use a variety of Internet-based services to accomplish many of my computing tasks. Even more than that, I pay for many of them!
Here's a list of cloud services I find valuable enough to actually spend money on them. This entry will likely get updated as I find new services and/or remember any of them that I've forgotten.
File storage stuff
Dropbox: A service for syncing files between different computers and devices transparently, through the concept of a single shared folder. It really does "just work". While there's a 2 GB free tier, I pay for the 100 GB tier because I never like throwing anything away.
Tarsnap: "Online backups for the truly paranoid", Tarsnap performs encrypted backups to Amazon's Simple Storage Service using extremely-strong, well-vetted and open-source encryption software. Fairly cheap, it's priced at "300 picodollars/byte-month" ($0.30/GB-month) after compression and deduplication. This is my off-site backup solution for my personal machine and random servers (on-site is an Apple Time Capsule, which is also incredibly convenient).
These ones are typically shared with my wife, Leigh.
Pandora: Basically, auto-generated Internet radio stations. I pay for the "Pandora One" service to skip the ads, and it's one of the best uses of $36/year I can think of. Maybe I'm lazy, but when I'm at work or concentrating on other things I'd rather not have to build a playlist, and I like getting new music. Pandora handles it for me.
Netflix: A huge archive of movies and TV shows, available all the time on most devices, for a low monthly fee? What's not to like?
Amazon Instant Video, Cloud Player, and Kindle: It doesn't quite qualify as a cloud service in my mind, because you have to buy specific items; but Amazon does make it easy to watch/listen/read your purchases on just about any device, just by signing in.
- LastPass: An encrypted password manager with browser integration, mobile clients, etc. I pay for LastPass because you can't get the mobile clients otherwise, and there are a few other value-adds. And right there, I never have to remember another password again. I really think that a password manager with good encryption, a long master password, and randomly-generated per-account passwords make for excellent security.
Honorable mentions: Free, but I'd totally pay if necessary
Bitbucket: A source-code hosting site with free private repositories, git and mercurial support, and some pretty decent wiki and issue-tracking features. I also like Github, and I use it for public projects because it has a larger mind-share in the software development community, but I generally prefer Bitbucket for personal stuff.
Google Apps: Including Mail, Docs, Calendar, and just about everything else under the sun. These tools are invaluable, and I love that I can set up Apps for a personal domain for free with little to no hassle. If I didn't live in a world dominated by Outlook and MS-Office, I'd do everything in Google's tools.