I logged out of iCloud on my iPhone and now it won't let me log back in

This answer offers an alternative to the one posted by user @Sacha on May 10th of 2018. For Apple users (like me) that cannot remove 2-factor authentication (because their account does not feature the link to do so), try this:

• Log into your Apple account from a computer (appleid.apple.com or icloud.com). That step will require you to use your 2-factor authentication, as expected, so have your phone handy.
• Scroll to the "devices" section. Verify that all shown devices are correct and not listed more than once. In my case, my iPhone was listed twice and that seemed to confuse the iCloud login algorithm. If that's the case, remove all duplicate(s) from the "devices" list so you only have a single phone listed there, and log out.
• From your phone, try logging in again to iCloud.
• If that still does not work, log into your Apple account from your computer again, and remove the device causing problem from the list completely. Log out. Now you have a clean slate.
• Grab your phone, and try logging in again to iCloud. This time, you may need to force the 2-factor code to be sent by SMS, instead of iMessages (since you might no longer have access to the iMessage app).
• Hopefully, this time around, your iPhone will complete the login process into iCloud and you will see your device on the Apple account page next time you access it.

This happened to me also. I somehow found it that my two factor authentication had been turned on.

Disabling Two-Factor Authentication on Apple ID 1.Open any web browser on any computer and go to icloud.com or appleid.apple.com 2.Log in to the Apple ID you want to disable two factor authentication for, you may need to use two factor auth to gain access to the account 3.Go to the “Security” section of account settings and choose “Edit” 4.Locate the “Two-Factor Authentication” section where it says the feature is ON, and click the link to “Turn Off Two-Factor Authentication” 5.Create new security questions to assign to the Apple ID, these are used in place of two-factor auth codes 6.When finished disabling two-factor auth you can log out of the Apple ID management website

Once two-factor authentication is disabled, you can log in to Apple ID from anywhere, whether on the web, iOS, iPhone, iPad, Mac, anywhere, with only a password again, you won’t need to double authenticate with a trusted device code.

Whether or not to use two-factor authentication is largely a matter of personal preference, there is no right or wrong answer here. Remember, you can always turn on two-factor auth for Apple ID again if you decide to use the service again at a later date. Either way, be sure to use a strong password.

Yet another answer, more simple than the other ones posted here, that I also tested successfully on my (now pretty old) iPhone 5C device to solve this aborted iCloud login problem:

• Grab your phone and go to "Settings" > "iTunes & App Stores"
• Login from there using your Apple ID, password (and 2-factor authentication if applicable)
• Wait a bit (it may take a few seconds for Apple to resolve multiple logins).
• Scroll back up to "Settings": you should now see your name and picture ID, indicating that you're also logged to iCloud. If not, stop the "Settings" app and reopen it to force it to refresh.

That is just an oblique way to log in when trying directly via iCloud does not work.