With some work, it’s possible to turn a modern Android smartphone into a Google-less, completely open device. FOSS stands for “free and open source software,” and when we say “free” we don’t mean free of cost, but free of restrictions. The SoC in the Nexus 4 is made by Qualcomm, and many of the drivers for it are closed source. The CyanogenMod repository has a list of closed source drivers in each device branch called “proprietary-blobs.txt.”
CyanogenMod takes AOSP, adds a bunch of handy enhancements, and ports it to tons of devices. There is a project out there called “Replicant,” which is based on CyanogenMod and aims to build a completely open source version of Android. There are only about 1,000 apps in F-Droid.
And in the end, the hunt for diverse apps such as word processor, maps or calendar happens to be a real hazard.