Phone keeps asking me to update Google play services but it won't let me

[AC Question]

When I open certain apps it says this app won't run unless you update Google play services. When I click on it to do so it sends me to a series of checking info.. pages to add Google account and I enter my email, hit next, enter password, hit next, then accept privacy terms and conditions, hit next, then it brings me back to add account page where I enter my email! Please help!