To finish out our hypothetical unofficial Android phone that has absolutely no apps by Google:
Amazon Appstore: The most obvious replacement for the Google Play store would be Amazon’s store for Android apps.
Cortana: Microsoft says it will bring Cortana to Android sometime after the release of Windows 10. This will put it in direct competition with Google Now.
Next Lock Screen: Microsoft developed this lock screen app as an alternative to Android’s default. Next is surprisingly pretty good.
Z Launcher: Nokia created this to help make launching apps faster, and with the ability to do so through gesture drawing.