Eagle's PCB ratsnest command reports 3 airwires, but they are not visible

There is a very useful eagle ULP called "zoom-unrouted" which if you haven't already got it, you can download here.

Save it in the ULP folder and then run it (run zoom-unrouted). Eagle will then zoom in to the first airwire.

Check for vertical (single pixel) airwires caused by ground plane islands, and for tiny gaps at the end of traces.

I can see from your screenshot there is one airwire between pins 7 and 10 of IC2.

On an unrelated point, make sure the "Isolate" value on your polygons is non-zero (it looks to be set to 0 to me) otherwise you will get Gerber files which are solid copper over the entire board (everything shorted out).

Also run the DRC with settings relevant to your PCB house as there appears to be quite a few places where there may be either shorts or clearance errors.

There's also an awful lot of redundant routing on the OUTPUT_LEFT and OUTPUT_RIGHT traces - unless you are going for star connected, you might want to revisit the routing of those traces.

Deselect all layers except the 'Unrouted' Layer.

You shall see which connections are open.

This happens to me quite often and it is usually caused by a mismatch in grid sizes between components. You think you run a track to a component pad but unless it ends at just the right point, it's still considered disconnected. What I do in this case, since I can rarely add a track that short to jump the gap or recreate the old grid, is use the autorouter. This is the ONLY time I use the autorouter. Just do a quick run and have it fill in whatever is left (don't have it redo the entire board) and it should make the missing connections.