Hello! I have a toilet that is leaking water in a rather unusual fashion. When the water is turned on it will very slowly leak until the float dropping causes it to refill. This happens only about once every 15 minutes. However, once the water is turned off it no longer leaks. I decided to let it sit overnight and the next morning the water level was still at the same place it was the night before. I don't see any water on the floor and the outside of the toilet appears dry. I don't know much about toilets so this has me stumped. Any ideas? Many thanks in advance.

How far down the overflow tube is the refill hose?

The water level dropping in the tank indicates that it is leaking out. The most likely sources are a leaking flapper, cracked/deformed flush valve, or a cracked rubber washer under a tank to bowl bolt. Your experiment of shutting the water off doesn't mean a lot; if it was draining from the tank into your house water supply, the water level would have to be above the top of the fill valve and under the top of the overflow tube while having a bad fill valve-- and the pressure in the house would have to be under the 12 inches of water column of the tank, but only for 14.5 minutes out of 15. (If this is happening you need to both buy a lottery ticket, and get your faulty well switch and pressure tank fixed, and find where the rest of the water is leaking)
Now, at this point I suggest either replacing the flapper in the toilet, or adding dye to the tank to see if it is leaking from the tank to the bowl, but refilling every 15 minutes is a good indicator already. Just to make sure that water isn't siphoning from the tank to the bowl from the fill valve refill tube, make sure that the little tube coming from the fill valve doesn't go down the overflow tube more than about a half inch, or use an air gap.
Use this link to our sponsor and check out the top picture for the component names, and specifically go to step 11 to see how the air gap is usually made.[www.plumbingsupply.com]
After any adjustments are made, you should have a space as is shown below, and your angle adapter which maintains that 1" air gap