The role of router as you find it on the networking diagrams doesn't do that sort of higher-layer caching. It just forwards packets to their next local network, rewriting the layer two data as needed.
Where it gets confusing is when the routing function is bundled into something else like a firewall. These days "firewall" can include everything from stateless port block/allow statements at the most basic, to application-level content filtering (smart enough to notice that someone is trying to mail company secrets to a competitor). The most featured of these devices aren't called Firewalls, they're called "Security Gateways" (at least this year) and 'firewall' is but one feature on the list.
These Gateways can very well cache content passing through them. Their whole point is to automate content inspection to a very detailed level to prevent key data going missing, and preventing users from accessing bad content.
Security Gateways, which have 'router' among their long feature lists, are typically deployed at the organizational edge. Because of how they work, they need to be placed in the network path of the interesting traffic, which is why they're typically the first or second device on the network after the ISP network-cable.
Where these gateways are NOT deployed are on ISP internal routing networks. Some ISPs offer these features as an extra service you pay for, so you'll know if it is happening. For ISPs, all they care about is Layer 3 transit unless you have a separate agreement in place where they care about higher level stuff too. Doing 'routine maintenance' does not expose reassembled client data to their technicians; if they're doing packet traces to isolate a problem client data may be in the capture, but they would have to go through significant, purposeful effort to turn that data into files/email.