Assigning a gameobject (not a prefab) to a script in the inspector

It seems like there is a restriction in Unity that prevents me from assigning a scene game object to a public script variable inside the script's inspector. Unity only allows me to assign things that are in my Assets folder, but I cannot assign anything that is in the scene. I assume this is because scripts are supposed to be universal, but instanced game objects are per-scene. Is there any way, then, to create a game object (not a prefab) that exists in all scenes?

Otherwise, it appears that my only other option is to assign game object manually to each and every GO the script is attached to, which is certainly doable but tedious, or use a private variable instead and assign it using GameObject.Find(). Is that correct?