How do I power/wire something that is rotating constantly WITHOUT extra batteries?
You are looking for a slip ring. This is a device designed to do exactly what you describe - transmit power and/or signals to a rotating object.  Generally, they work by having a rotating circular contact on one side and a spring-loaded pin which pushes against it on the other. 

They are not super reliable long term or in harsh conditions, but should be fine for your project.  More expensive ones are generally better and last longer.  The other alternative is inductive power transfer/signalling, but this will be harder to get right first time, so might not be ideal for a hobby project.