Does a network need a new router for every change to the network medium?

I feel like if I don't explain this right, my question may not be understood but here goes. I'm in a intermediate level networking class at my university and we're learning about the OSI Data Link Layer. The Cisco textbook we're using, "Introduction to Networks" v7.02, describes the duty of a router to be (in terms of the data link layer) accepting a frame, de-encapsulating it, re-encapsulating it, then sending it on the the next segment of the network in a format its medium understands. My question then is this - must a network have a new router for every change to the network medium? As an example, if I want to send data to someone, and between me and them the network changes from, say, my Wi-Fi via cable internet, to my home city's copper wire WAN, to a long-range fiber optic wire, back to the receiver's home city's copper WAN, and finally to their own dial-up system, is there a router at each medium change that handles the handoff of the data as it travels?