I tried to place a repeater and connect it with redstone in a loop around the repeater, then I gave it the redstone signal. When the button came back up, the loop stayed lit. I knew this was going to happen, then I connected the loop to a dispenser, and when I put arrows in it, it didn't fire! I had done all of this on my iPad. Why didn't it work?
It seems you did a small mistake:
But, when you place a button next to any of the four redstone wires, and press it, both repeaters will power, and it won't create a functional clock (loop), like this:
Now, when you get this, in order to make it work, you have to quicky destroy and replace a redstone wire of your choice. An easy way to do this is to link it to a redstone block one block below the redstone wire and place a redstone torch on it. This will turn the circuit on for 1 tick before the torch is powered down. You can also set a delay to the repeaters (the farther the red buds on the repeater from each other, the bigger the delay), so you can replace the redstone easily. The final clock (loop) should look like this:
This is at minimum repeater delay, the clock can go up to 4x slower.