Off the top of my head I can give examples of Natural History museum and technical museum. Both are 8€ for an adult and are on par with their peers in London (that I've also visited).
Berlin Cathedral is 7€, Pergamon museums is 12€, Altes Museum is 10€, Alte Nationalgalerie is 10€ – there are 5 museums on the island of museum altogether, a ticket to visit them all is €18. Not free but not crazy expensive.
Prices correct as of September 2016.