Hello Henri,
Sorry for the confusions by my colleague earlier as I believe he may have misunderstood your question.
It is DEFINITELY possible to insert images with TinyMCE and this is the standard behavior of Joomla's TINYMCE plugin. Take a look at my screen shot here,
http://take.ms/zlrHn
Alternatively, you could use the insert media button in EasyBlog,
http://take.ms/Ooi5u
By the way, it's most likely that your TINYMCE is configured to be on simple mode. You need to change it to run on "Advanced" or "Extended" mode as you can see here,
http://take.ms/TfMc8
To configure the plugin, go to Extensions > Plugins . Search for TinyMCE. Click on it