NAS apps sound interesting, but not what I'm looking for. Router apps have untapped potential because routers are relatively inexpensive, always on, and have a real Internet address, unlike the rest of the home/office network.
What kind of apps would developers create? All kinds! Some would succeed and some would fail, just like phone apps. The point is to unleash developer creativity. Today, nearly all Internet traffic is routed to Netflix, Google, Amazon, and Facebook. Over the next few years, consumers will buy millions upon millions of upgraded routers. I think they would like to buy routers that do more for them than send all their traffic to a tiny number of huge companies.