How can you tell how many layers a PCB has?

As far as non-destructive methods, you may be able to shine a bright light into the edge and through a corner see the copper planes. Signal traces may be hiding though, and this only works if the copper comes fairly close to the edge, which it may not.

Using a bright light, it's easy to see if a board has inner layers even if it doesn't have blind vias. Find some place on the board where there aren't traces/planes on the visible, outer layers and see if you can see light through it. If it's blocked in some places, that's probably copper. Even without that, most multi-layer boards I've dealt with using the standard green LPI have a darker appearance than their 2-layer counterparts.

If the board designer had space to kill or the manufacturing engineer wanted for some reason (not sure why), I've also seen boards with a "stair-step"; each layer labeled with a copper number and cutouts in the other layer to be able to see it. This might be unique to my old company though, as our layout tech did all sorts of strange things.

Cut the board in half and inspect the copper.