Here is some facts I did not provide when we started our conversation.
First of all this is the site URL settings page hidden link. And this are the settings in the WPML -> Languages section hidden link. We want this settings to have /el in all default language pages. After just saving the settings in the permalinks section, without making any changes, the issue was resolved. We have not made any root page, have not setted the home page template to be the redirect one that you have in your documentation and we have not tried what you suggested because it will cause more redirects and the page will take much longer to load.
We do not have activated any cache system, nor made any changes in the .htaccess file or added snippets in functions.php.
This is the default language home page URL: hidden link
As you can understand when someone presses the default site URL he gets redirected to the Greek homepage. It seams to work now and pretty fast but we can not locate the cause of the problem with ERR_TOO_MANY_REDIRECTS. Is wpml redirecting the site base URL to the Greek homepage?
As you can understand we can not save the permalinks settings every time there is a problem. We need a solution as soon as possible.
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We were unable to reproduce the issue in a fresh or minimal installation on our end. Typically, creating a root page should not automatically redirect from the site URL (hidden link) to the language directory URL (hidden link). It should remain at hidden link and only redirect after a redirection option has been set.
Therefore please allow me to make a copy of your site using the Duplicator plugin (https://wordpress.org/plugins/duplicator/) for debugging the issue on my local server. So we can troubleshoot the bug without affecting your live site.