How much does it cost to charge a laptop battery (1 cycle)?
You could bound it with the wattage rating of the charger and the time connected, try to calculate it from the battery capacity and an efficiency guess, measure it by plugging it in though a "kill-a-watt" meter (which may actually have an integrate-over-time mode)... 

But, while you didn't include it in your question, I think you might also want to take into account consuming the usable lifecycle of the battery; it's at least worth estimating if that might cost more than the energy to charge it.  

So - the battery out of my compact laptop is rated for about 60 watt hours.

Let's assume we're in NYC where electricity is 20 cents per killowatt hour, and for a worst case that the charger is 50% efficient.   .06 KWH * 20 cents is 1.2 cents, double for the 50% efficiency is 2.4 cents.

I don't know what the lifetime of the battery is, but 1000 full cycles sounds rather optimistic - at that rate, the lifetime energy cost of charging it would be $24... and I'm pretty sure a replacement battery costs more than that.

Also note that you normally wouldn't fully discharge the battery before recharging it.  Different battery chemistries and even manufacturers may have different ideas of between what charge states the rated capacity should be measured.