You could store it in a text file, which might be marginally easier than storing in a database if you’re not familiar with databases. Ajax would be a good thing - use it to trap the button press, call some code on the server to update the stored number, and then update the display to show the new value (as others might click it at the same time).
OMG NO. Only if you want to get into real trouble and do not need any valid number over time. Plain text files are a mess when it comes to concurrent writes. If you want results you can rely on, use a database with ACID compatibility, at least SQLite, it’s build in and you get a counter with a few lines of code.