> I would like to know what happens when you opt for private health insurance but then require a treatment that the insurance won't cover
They are required to provide a certain level of care to be allowed to provide the required insurance. There are things the GKV ("Gesetzliche Krankenversicherung" - the state organized insurance) provides which the PKVs ("Private Krankenversicherungen" - (partially) profit oriented insurances) don't have to, but I know of none that are life critical. Many of the things not necessarily payed by PKVs are in the somewhat affordable range.