You can be pretty certain that any free software apps found in F-Droid will not require proprietary Google APIs. Every other app in the Play Store can be assumed to require them.
I don't know, but please tell me if you find one. I have F-Droid and Play on my phone, and disabled Google Play Services to see how many of the apps I have installed and use continued to work. Many gave me "must update Play Services" dialogs, some just crashed outright. I've been trying to transition to open apps or apps available outside the Play Store, but unfortunately, Skype is one need I can't work around right now.
