I have installed an app which apparently requires a newer version of Google Play Services. How can I upgrade these services to a newer version without having a Google account?
This is on Android 4.4.2.
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You will need to goto a 3rd party website that hosts Google apps and download the newest (for your device) version. The one I use is http://gappsandroid.blogspot.com/ , you can download EVERY single Official Google app.
You could use OpenGApps. I believe it is sanctioned by Google and allows you to easily select a version for your device architecture and Android version, customizing the package by which apps/services you want.
Once downloaded, you can flash the package via recovery. It is also possible to extract the apps for a manual upgrade; see my answer here for details.