I had something of a "Galilean moment" on a flight from California to Philadelphia once. For the first and only time in my life, I saw another commercial airplane flying outside the window of an aircraft I was traveling in. The sky was opalescent blue, and we flew above the cloud layer, which meant there were no other background objects with which to compare myself and the other plane - a situation which only typically exists in the relativist's thought experiment.
For argument's sake, our plane was traveling East, and other plane, South, and we shared the same local atmosphere. To my eye, looking out our portside, the other plane appeared to be traveling as indicated by the arrow in the left half of the below diagram. It's a strange thing, to see a plane flying at an angle that is not aligned with its axis of symmetry. An earth-bound observer, of course, would instead see the arrangement on the right of the diagram (the one which agrees better with the principles of flight), assuming no observable wind effects.
Several seemingly trivial thoughts occurred to me. The first is that, irrespective of the earth (which I could not see), I knew that a third frame must exist. The second is that I only knew that a third frame must exist because of prior information that I could assume is true, namely, that to fly, the thing that is a plane must be propelled along its axis of symmetry if in Earth's atmosphere. And finally, it was critical that a plane has a single axis of symmetry. Were it a sphere, I would not be able to distinguish what the craft's direction of propulsion ought to be.
Hidden among these realizations is my role as an observer. Rather than coordinating flight patterns in a thought experiment as an omniscient observer (one who can boost between frames), I was in fact on one of the planes trying to piece together what I could understand only with my visual observations of the scene. And from this perspective, with my prior knowledge about air travel, I could infer the existence of a third, Earth-bound frame.
All this highlights the importance of checking our own assumptions. The intuition at the root of the observations here likely contributed to physicists' insistence on the existence of the luminiferous aether. If only we were also privy to first-hand special relativistic observations, as in Gamow's cyclist thought experiment.