I've a budget of ~800€, and want at least:
* an SSD * 12/16GB of RAM * a decent CPU
Any recommandation? (ps: I don't need a screen, only the computer)
I've set up several people I know with a Raspberry Pi 3 and they're all happy with them. It comes with Chrome and Libre Office so that's covered for you.
You can get one (if you hurry) with a 375gb hard drive, power supply, and a very nice case and cables from WDLabs for around $100 and set it up to boot directly from the HD so you don't have to worry about SD card corruption.
I tweaked the desktop UI a bit on those I've set up to hide most of the Pi's educational and programming tools to make it easier for them to learn and navigate but everyone I've set up had no problem making the transition from Windows to the "PIXEL" desktop the RPi team has developed.
I'm still using an old "Mac Mini" but when it dies I will make the switch too. I mostly code web apps and don't need high end graphics or tons of memory and the speed difference in loading apps and running them isn't enough to think much about.
With the money you'd save you could take you mom out for a nice dinner once a month for the rest of the year or, if she doesn't like it, you can still get her a decent Win box and make something with the Pi for yourself, (like this beer brewing setup : http://web.craftbeerpi.com ;)