I changed the password on my Google account a week ago, and ever since then I've been getting notifications on my Nexus 4, requesting permission for my account. When I press it, nothing happens, but Google Play Services appears in the recent apps menu, and nothing happens when I press it.
I had the same problem, it's a bug that appears occasionally when you change your Google account password, or when you switch to the two step verification. Some apps that are using your Google account can't synchronize the new password and this error appears.
In my case, the guilty was the app "My Tracks" (from Google) (https://play.google.com/store/apps/details?id=com.google.android.maps.mytracks&hl=en). Going to "Settings > Apps > My Tracks", claning data and cache and relaunching the app solved the problem.
If you don't have "My Tracks" app installed then try this:
Go to "Settings > Accounts > your Google account". Start unchecking one of the options and select "Sync now" (in the menu). Continue unchecking the rest of the options one by one and doing "Sync now" until the synchronizing error stops appearing. At this time, go to "Settings > Apps", search for the last app you unchecked, clear its data and cache and relaunch it (the app may ask for permission, you have to allow it). Then return to "Settings > Accounts > your Google account" and check all the options and do "Sync now". If the error continue appearing it means that another app is failing, so repeat the operation to detect and correct it.