I may have been wrong with the sizes. The hole diameter may be 4". I'll have to confirm this.
RFID is a very interesting option. It would allow me to identify which player's washer went through, which would be great. But I would need to rule out washers sitting on top of the board (that did not go in the hole). And I would worry about the constant throwing and hitting of the board wrecking the RFID chip.
I was just kidding about the RFID, but it seems to have the unique advantage that you could let several people throw at the same time and still keep track of it all. I have no idea how durable the tags are, but generally speaking if there is a demand, then there is a part that will work. You might want to switch away from metal washers to something a little less troublesome to RF, like plastic or rubber with the tag embedded inside.
@crossroads: I would only use the reflection sensors along with the direct beam sensors, sort of a backup way to catch anything trying to slip through the beams. All in all, a laser diode is probably the best bang for the buck solution so far and a little bit of water vapor or smoke combined with a visible red or green diode wouldn't hurt the wow factor any.