What should I do with a 13v 4 amp transformer?

First thing that comes to my mind is a car battery trickle charger. You’ll need regulators and filters to make it work. It’s probably not worth the effort since you can buy a complete unit for $15.

That’s a lot of amps to use anything except a larger car or motorcycle battery.

Things designed for 12–13V will use only the current they need, so should be OK to use this with any 12–13V device. In fact, you can connect a bunch of 12V devices in parallel until you get close to 4 amps total. If something shorts, though, 4 amps is a lot of current.

@RocketGuy You can burn up a lead cell battery by putting many many amps into it because it will accept the amps. There is not demand limiter in a battery being charged.