The problem with "Update never finish" was:

When there was an update for Messenger from ver. 13.0.1 to ver.13.0.2 the upgrade finished OK but after upgrade "Standard users" got 'Error 500' when entering the Homepage where Shoutbox is a part of the page. I thought that the problem was new, upgraded Messenger so I tried to restore old ver.13.0.1 which I did but I made a mistake and instead of removing 'messenger module folder from '/modules/boonex/' I just renamed it to "messenger_ver.13.0.2" and that was the reason why I had that problems with upgrade. But even after I solved that problem, Standard users still had the Fatal Error 500" - the problem was in Profiler which was enabled. After I disabled "Profiler module" Standard users can access Home page and Shoutbox.

I would like to say thanks to @Alexey for his help around this issue.