A Mobile that sits on a table and has several stages on top that turn and tip freely, I've built them in the form of a Christmas tree, or multiple 3-D tetrahedral stars. Many other forms are possible. It's done by having parts of each stage lower than the pivot point, and loaded with weights, and the parts above the pivot points as light as possible. The center of gravity (including whatever stages may be balancing on top of said stage) must be below the pivot point.
The other two are "vehicles".
The first one is a "cart" with two side-by-side bicycle wheels, which keeps itself in balance with no other ground contact. Here the axles of the wheels represent the "pivot line", and again, the center of gravity of the bed of the cart and it's load must be below the pivot line. I got the idea while making a cart to pull behind my bicycle, and imagined it with the external form of a large goose with cardboard "legs" on the wheels. It can be battery-powered (with the batteries probably also used as ballast), and with a fairly simple circuit (consisting of photocells with lenses, diodes, amplfiers, relays and motors) will follow a light source, staying at a set distance.
Recently I advanced to figuring out how to make it balance on one wheel, without a gyroscope. In this case the wheel appears solid and the ballast is inside the wheel. It could be battery powered, but a steering mechanism would be difficult. It might be in the form of a "BC" cartoon figure riding his stone wheel unicycle.