In principle, all orbits in the universe should progressively decay due to the emission of gravitational waves. However, does this always happen? Are there any kind of orbits that do not decay as they do not emit gravitational waves? Perhaps if the orbit is perfectly axially symmetric with an even distribution of mass?...
All orbits as you would normally think of them emit gravitational waves. You mention an axially symmetric orbit. A hypothetical rotating rigid hoop would not, since the distribution of mass is not changing over time.
The difference between the gravitational waves produced by neutron stars or black holes vs planets or stars orbiting the galactic center is just a huge difference in degree. From what I have read about the long term future of the universe, on trillion-year timescales, Newtonian interactions between planets lead to them getting ejected before orbits can decay from gravitational waves. (This is a statistical analysis; orbits cannot be forecast nearly that far in detail.)
On the galactic scale, it remains true that energy loss due to gravitational waves is such a weak phenomenon that other effects will determine the ultimate fate of the (by then mostly extinguished) stars. See this link (from the comments - thanks to Sten).
In a sense it is almost true that ordinary orbiting planets, or stars orbiting galactic centers, don't emit gravity waves because the waves they emit are so weak. The energy in the waves has to do with the masses of the bodies and the period of revolution, and longer periods mean less energy. The stately progression of stars around the galactic center are not going to produce very energetic gravity waves.