Arch Linux might be what you're looking for, if you're willing to take the plunge. You get a bare bones machine when you install it, and you make your own desktop experience. You can install GNOME or KDE, or something more lightweight like openbox, or a tiling wm like i3 or xmonad.
What you want, though, is something like PPAs. You're looking for the Arch User Repository [1]. You can install some helper like Yaourt [2] and run something like `yaourt -S leagueoflegends` to get user-contributed packages.
The other thing that really does it for me with Arch is the excellent wiki, which documents nearly everything you'd ever want to do with Arch Linux.
[1] http://aur.archlinux.org/
[2] https://wiki.archlinux.org/index.php/Yaourt