Tested on Chrome reported version #62.0.3178.0 dev and latest Chrome stable #60.3112.101 and unable to reproduce the issue on Mac 10.12.6 and Windows 7. Attached the screencasts link for reference. (https://drive.google.com/drive/folders/0B91MtOa4hcAbX2xLSUlUREcwa3c?usp=sharing)

Tried by checking and unchecking the checkbox “Also remove your existing data from this device” while logging out of the logged in users.

@zhaoz --thanks for posting this issue.  Could you please let us know while logging out of Chrome, did you click on the check box “Also remove your existing data from this device”.  You can also try by removing the extensions and creating a new profile to verify if the issue still persists.

Please let us know if we have missed anything.

Thanks in advance.

Unfortunately, I no longer have this mac as I have returned it for an upgrade.

I don't remember there being a checkbox.

That said, I did try to create a new profile. This was done through the mac menu because the chrome profile switcher would not work. Creating the new profile would open a new window that was always underneath the window asking me to sign in. Any clicks in that window would be interrupted by the broken profile window. 
I would quit chrome completely, and try opening it again and it would always start with the same broken profile. I could have probably tried starting it with some command line flag in the new profile but its too late for me to try now.

I understand that this may be a unique corner case and hard to repro. Unfortunately, I do not have the laptop on hand anymore for further investigation.

I really should have taken a video or something.

Thank you for providing more feedback. Adding requester "pnangunoori@chromium.org" to the cc list and removing "Needs-Feedback" label.

For more details visit https://www.chromium.org/issue-tracking/autotriage - Your friendly Sheriffbot

Removing 'Needs-Bisect' label as the issue is not reproduced at our end.

@zhaoz -- Thanks for giving reply. Could you please let us know if the issue is reproduced or noticed again on your machine. And kindly provide video, this would help us in reproducing and triaiging the issue further.

Thanks in advance.

--Chrome Identity automated triaging--

This bug is Unconfirmed and has gone two weeks without any activity, so it is being closed as WontFix. Please re-open if this is still a valid and reproducible bug or feature request and mark it as Available. Please see https://goo.gl/78kbny for more details. Please remove the Services>SignIn or UI>Browser>Profiles components if this bug isn't related to Chrome Identity.

For more details visit https://www.chromium.org/issue-tracking/autotriage - Your friendly Sheriffbot

Sounds like a misbehaving extension. If you have installed an extension that requires access to your Google Account that's signed into Chrome to work, it can continually pop up the signin prompt. See Issue 695963 for more background.