you could just remember it yourself! creating the ultimate live web 2.0, where once a user enters a username and password, you actually have to verify it at that exact moment or else they can't login! then you could hire midgets and have them remember and rebuild profiles for people and unicorns and...
anyways, i thought most (simple) desktop apps store data in text files? (perhaps encrypted?) it's definitely possible to store data in text files for web apps but seems inefficient for many reasons - speed and scalability being the first that come to mind. i've stored data in text files before for very simple apps that contained a couple of fields and had no sensitive information.
my personal suggestion: go with MySQL. it's free, well documented, efficient, scalable, etc. the list goes on...
The filesystem doesn't handle concurrency and atomicity of some operations well, while in web apps it is absolutely necessary - you serve multiple users concurrently and you will inevitably have some shared data.
I tried to avoid databases in my web projects in the past and ended up having quite complicated concurrency infrastructures, based on shared memory and the filesystem. I can't say I was happy with the complexity I got.
Databases, no matter how ugly, clumsy, handle it nicely. In a few cases I made a complete switch to DB and had my code reduced significantly.