Rpi's are great for "always-on" stuff, as mentioned already. This makes it great headless (no monitor) applications like: for network attached storage (NAS), transmission-daemon, mopidy, piratebox.
I've tried all of those. For the nas, I went bare-bones with samba. There's good instructions for samba at https://wiki.debian.org/SambaServerSimple
. This is probably the thing I use my rpi3 for the most. I don't back up all my files to it, but I back up my 'Documents/computer' directory (where I keep all my configs/how-to's/etc that I make) from my Desktop. The nas makes a great go-between for other computers that aren't on. So if I want to save something to my main (office) computer, I put it in my samba share in a directory called 'transfer-office'. I have another called 'transfer-phone'. On android I have the ghost-commander file browser with the samba plugin.
Transmission-daemon serves up a web-interface so I can download a linux distro and keep seeding it until I've got a high share ratio or it's not being 'accessed'. This is great for starting downloads and turning off all other computers. There's good instructions at https://www.robertsetiadi.com/installin ... pberry-pi/
. I got stuck with the config because I was editing the line rpc-host-whitelist (or something like that) when I should have been editing rpc-whitelist.
Mopidy is good if you've got some speakers hooked up to the pi. Not the greatest sound without a hardware addon, though. Control mopidy from any browser on your lan.
Piratebox I played with and it was cool.
Other self-hosted things with web-servers I'd like to try: calibre, etherpad (apt-cache search etherpad gives output of 'node-channels - event channels in node.js'), sonerezh, syncthing, tinytinyrss.
If you like to tinker, get a pi for sure.