To get the best experience, do not try to hunt for the absolute "newest" APK. Seek out the last stable release confirmed for Android 4.4 (specifically Play Store version 5.10.30 or 6.0.5 ). While the digital world has moved on to Android 14 and 15, this specific APK remains the gold standard for keeping the venerable Android 4.4 alive.
While F-Droid doesn’t host Play Store, it hosts "Aurora Store"—a lightweight client that acts as a proxy to Google Play. For Android 4.4.4, Aurora Store v3.x is excellent. google play store apk android 44 4 best high quality