Windows 7 64-bit

It means what it says. Please check the login, password and permissions.

But I'm running the command using
-u leutek
why is it telling me that user root has no access permission?

The only thing that comes to mind at the moment is that you have some kind of pluggable authentication. Try to run select user, host, plugin from mysql.user to see if there is anything suspicious.

You've set the affected version to 5.5.41, is it really the case?

I have checked again, it's version 5.5.47

here is the output of select user, host, plugin from mysql.user

there is no root user

eidbadr,
I assume it answers your question, right? Somebody or something dropped the root, you should re-create it.

yes the root is dropped, and I'm using another user to perform the operation. Is that not possible? Should the root be available to complete the opeartion?

Sorry, my bad, I forgot the nature of the problem – you are running upgrade under account leutek, but you get an error message about not being able to connect as root@'::1'.
It's bizarre.
Did you try to connect by MySQL client? Are you getting the same error?
Also, since you have this output from a GUI (HeidiSQL?), apparently you can connect via it. Are you using the same leutek account? Could you please run SELECT CURRENT_USER(), USER() and SHOW GRANTS there?

eidbadr, please see the question above.
I don't want to close the issue as not repeatable, because obviously something weird is going on here, and it's worth digging into.

@Elena Stepanova

I don't get the error in 5.5.48 anymore.

Connecting via HeidiSQL and MySQLWorkbench using user leutek have always worked.

output of SELECT CURRENT_USER(), USER()

output of SHOW GRANTS

'GRANT ALL PRIVILEGES ON *.* TO \'leutek\'@\'%\' IDENTIFIED BY PASSWORD \'*EC9DC28A0F0B3AF8B1C752B6342A20E4EE272485\' WITH GRANT OPTION'

I've got an idea what was going on there, but since it stopped happening for you, there is no way to check it.
I can reproduce the problem almost precisely:

In this case I used U+00A0 symbol (non-breaking space) instead of all spaces on the command line except for the last one. Apparently it does not allow to read the options, so tools use default values; for mysql_upgrade the default user is root. I suppose it can be so not only with this particular symbol, but with some other ones as well. I don't know where they could have come from – maybe unusual keyboard layout, or unfortunate copy-paste, or anything like that.

Do you think it could have happened in your case?

Please comment to re-open if you have further information on the issue.