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.

An alternative to slip rings is the rotary transformer. This consists of two cup-shaped cores that face each other, with the windings inside the cups. If you drive the primary at a high frequency (we used 25 kHz), the cups can be ferrite and the whole thing can be quite compact (we did about 100W in a unit that was about 1" thick overall and about 3" in diameter).

How about using two rings of ball bearings and use each ring as the contact. No need to add pins and contacts...

Just to add, after taking note of the comments, that I was thinking of a low voltage supply here 5v or 12v just for leds or somesuch - I was not intending to suggest a 110VAC or 230VAC solution.