Unable to reproduce the issue on Windows 7, Linux Ubuntu 14.04 & Mac 10.12.4 using chrome reported version-55.0.2859.0 , stable-58.0.3029.81 & canary-60.0.3080.0 as per steps mentioned in comment#0.
Observed no Translation popup appears on every reload once we select "never translate this site" this option for a website. Tested below sample URL's:
-> https://id.yahoo.com/
-> https://www.yahoo.co.jp/

Please check the issue by upgrading chrome to latest stable/Canary & let us know your observations on the same.
Please find the attached screencast on Linux for reference.
Thank you..!!

Deleted: 663934.mp4 5.9 MB

	663934.mp4 5.9 MB View Download

Could you please confirm with file:// URLs - as the report specifies?

Could you please provide us sample file:// URLs to triage this issue further.

Thanks in advance.

I guess this is related to the fact that origin for the file scheme is defined as null, and Chrome fails to store per-origin settings as we intentionally do for content permission settings.

This issue is reproducible with any file that requires translation.

Yana, we need a product decision here. One option is to disable "Never translate this site" where we don't have a valid origin. WDYT?

It's not the ideal solution from a user perspective since what the user really wants is no translations on that specific file:// url but since it sound like that's not possible your proposed solution is definitely better than expecting the translation to not show up and being unpleasantly surprised. So let's go ahead with it.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/c452aca7b17c8938d24dbc0a17490eb01b9cb065

commit c452aca7b17c8938d24dbc0a17490eb01b9cb065
Author: Jon Napper <napper@chromium.org>
Date: Tue Oct 17 23:51:43 2017

Only show "Never translate this site" when site can be blacklisted.

The option to "Never translate this site" is always shown on the
translate dialog, even when the site cannot be blacklisted (for example
a file:/// URL). This patch changes the translate options combon so
"Never translate..." is only shown if it will have some effect.

Note that the 2016Q2 experimental UI has not been patched as the
currently plan is to not launch this UI.

Bug:  663934 
Change-Id: I64b80d830e124e9372906bd0c4711568c46ff636
Reviewed-on: https://chromium-review.googlesource.com/572475
Reviewed-by: Jia Meng <jiameng@chromium.org>
Commit-Queue: Jon Napper <napper@chromium.org>
Cr-Commit-Position: refs/heads/master@{#509607}
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/translate/translate_bubble_model.h
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/translate/translate_bubble_model_impl.cc
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/translate/translate_bubble_model_impl.h
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/views/translate/translate_bubble_view.cc
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/views/translate/translate_bubble_view.h
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/views/translate/translate_bubble_view_browsertest.cc
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/chrome/browser/ui/views/translate/translate_bubble_view_unittest.cc
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/components/translate/core/browser/translate_ui_delegate.cc
[modify] https://crrev.com/c452aca7b17c8938d24dbc0a17490eb01b9cb065/components/translate/core/browser/translate_ui_delegate.h