Hi Tommy,
The icon is in the RibbonWindow template with name "PART_ApplicationIcon". In an inherited RibbonWindow class you might be able to get it via GetTemplateChild and then manually hide it.
Are you using the older orb style of application menu? If so I would suggest you use the scenic style of ribbon instead if you want to use multiple ribbons. With that style, I was able to get things looking ok with this basic XAML:
<ribbon:RibbonWindow x:Class="WpfApplication1.MainWindow" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml" xmlns:ribbon="http://schemas.actiprosoftware.com/winfx/xaml/ribbon" Title="MainWindow" Height="350" Width="525" Icon="ActiproIcon.ico"><StackPanel><ribbon:Ribbon><ribbon:Tab Label="Home" /></ribbon:Ribbon><ribbon:Ribbon IsApplicationButtonVisible="False" QuickAccessToolBarLocation="None"><ribbon:Tab Label="Home2" /></ribbon:Ribbon></StackPanel></ribbon:RibbonWindow>