Both hosts and routers run TCP/IP protocol software. However, routers do not…

Concept: In a layered protocol stack the layers split into two groups by where they must be implemented. The lower, hop-by-hop layers move a packet across one physical link and decide where to send it next, so every device that handles the packet along the way must implement them. The upper layers are end-to-end: they exist only at the two communicating end systems, because only the originating and receiving programs create and interpret the data being carried.

In the five-layer TCP/IP numbering used by this question the stack is: Layer 1 Physical, Layer 2 Network Interface (data link), Layer 3 Internet, Layer 4 Transport, Layer 5 Application.

Applying it to a router, which is an intermediate forwarding device and not an end system:

1. A router terminates a physical link on each of its interfaces, so it must receive and transmit raw bits — it needs Layer 1 (Physical).

2. It must strip the incoming frame to recover the datagram and re-encapsulate that datagram in a new frame for the outgoing link, which may use a different link technology — it needs Layer 2 (Network Interface).

3. It reads the destination IP address, looks it up in its forwarding table, decrements TTL and forwards the datagram towards the next hop — it needs Layer 3 (Internet).

4. It is not an end point of the conversation, so it never has to rebuild the end-to-end byte stream (Layer 4, Transport) or interpret the message the two programs are exchanging (Layer 5, Application).

Cross-check by asking which headers the device touches: a router rewrites the Layer 2 frame header and updates the Layer 3 IP header, but the transport and application bytes travel through it untouched as opaque payload. A layer whose header the device never inspects or modifies is a layer whose protocol software it does not need for forwarding.

A router may of course run application programs of its own, such as an SNMP agent, a Telnet or SSH management console or a routing daemon, but those manage the router itself; they are not required to forward transit traffic, which is what this question asks about.

Therefore the layer for which protocol software is not needed by a router is Layer 5 (Application).