i'm looking for a possibility in QGIS to identify the intersecting parts in one single polygon layer with ten features, where at least four polygons overlap.
You can use the "TopolyChecker" Plugin.
When you activate it click on Configure and use the layer with the overlapping features and as rule "must not overlap". Then add the rule.
Afterwards click on validate all. You will get a list of all geometries that are overlapping other geometries in the same layer.
]1 But how you can add a rule with 4 unfortunately I have no idea if this is possible with the normal qgis tools.
I would use Join Attributes By Location, located in the Vector menu, under the category Data Management Tools.
Choose the option Take Summary Of Intersection Features, and check the box sum. This summarises all numerical attributes of the intersected polygons in new attributes (which you didn't ask for), but it also creates an attribute called Count, which lists how many features touched each intersected feature.
I just tested this with two polygon layers, and it worked fine.