Here is an excellent overview by”Learning SharePoint“on what the various top level elements on your SharePoint online are and how to hide them if you want to!
In short (mostly for my own reference):
<script language=”javascript”> _spBodyOnLoadFunctionNames.push(“HideBrandingsuite”); function HideBrandingsuite() { //hide the left upper "office365" logo document.getElementById('suiteBrandingBox').style.visibility = 'hidden'; //hide the top level links to outlook, sites, calendar, people, newsfeed,... document.getElementById('suiteLinksBox').style.visibility = 'hidden'; //hide the name, cog and help document.getElementById('suiteBarRight').style.visibility = 'hidden'; //hide share, follow and edit document.getElementById('RibbonContainer-TabRowRight').style.visibility = 'hidden'; } </script>
You can find much more information and code on the learningsharepoint.com website!