How does this app know I've used it before? There is an app that has a paid service. It allows you to use a free trial 3 times for 1 hour before it detects that you have used your 3 trials and blocks you. I tried various things, but each time it knows I have used my trials:


un- and re-installing
clearing app data
creating backups with titanium backup and restoring it
creating a new user account on my phone (android M), installing the app separately


How is this app detecting it? The only way I can reset the trials is by resetting my phone, but I can't keep doing this. Is there a way I can monitor the files the app creates/reads/writes?
Without knowing which app you are using, it's hard to provide feedback. If you have given file access to the app, it could have stored the trial usage somewhere on internal storage or on an SD card (if present). The best you can do is to search through files until you might find something.  

It could have also associated itself with your Google ID. Did you ever accept a dialog asking you to choose which Google account you want to use? If so, you can not use the same Google account for a trial on any phone, and would have to use a different account.  

That being said, if you have used your three trials, it seems that this is a product that you want to use, and it may be worth paying for. Keep in mind that a developer/company wrote the app, and may be providing infrastructure for its use, which costs them money. Sometimes, there are no free alternatives, and you'll have to pay to use a product or service.