My rather verbose two cents, for whatever that's actually worth.
I currently own over 200 pairs of headphones and earphones along with 15 different amps. If I could only own a single rig however, it would be a Stax flagship (SR-Omega, SR-007, or SR-009) and a high-end stat amp (DIYT2, BHSE, LL, or Electra). No question in my mind.
I've always approached this hobby from the mindset of a collector and archivist. To that end, the headphones themselves have been the object of my desire, both as a cultural artifact (their design, their context) and as different means of exploring music due to their various colorations.
If I were to have a single rig on the other hand, I would approach it from a completely different paradigm. The headphone I'd choose would be the one I felt best fulfilled its role as a headphone, not a cultural or personal artifact. For me this means the headphone that transmits music while not transmitting itself as much as possible, ie. the headphone that best gets out of the way and disappears (this includes wear comfort), the metaphoric window onto a recording. This isn't just a matter of frequency response however, but also listening involvement and spatial reproduction which can be highly personal factors. The Stax flagships when properly driven by a capable amp are the closest I've come to this through a pair of headphones as the vector.
While I personally think neutrality is a good thing, my definition of transparency has changed a bit over time. It's not necessarily a matter of neutrality, but rather it's defined in a negative sense by a lack of distraction. That's what I mean by "gets out of the way" ultimately. For some people, a neutral headphone might make its presence fully known by virtue of the fact that it's not making its presence know. Think deafening silence. A lack of coloration thus becomes a coloration. I think when one starts listening to the music and hearing past the headphones, then one has reached a state of transparency. In other words, for me transparency isn't so much an attribute as it is a process. One can listen past a pair of $20 earbuds and experience the music itself more than someone using music as a tool to feed a pair of $2k headphones. I don't think there's anything wrong with the latter however; it's merely a different set of needs and priorities.
For me personally, the SR-Omega, SR-007, and SR-009 have the least wrong with them and are thus the least distracting for me the most consistently. I can still enjoy music using a $20 pair of earbuds, but it's extremely circumstantial and takes a lot of effort. It's a classic case of ignorance accommodating bliss.
As an aside, I think there's a tendency for some head-fiers to fixate upon the "magical" quality of music. When we take part in head-fi, we enter into a symbolic order of established terminology and norms, and we're granted a language to use in speaking about our experiences and an audience for doing so. The catch is that we must distance ourselves from the immediacy of that experience to speak intelligibly about it. Our primordial enjoyment is now subjugated and contained within the order of audiophile culture. There is however a remainder left over, as it can't completely incapsulate that experience. It's felt in the discrepancies between impressions, the inadequacy of certain terms, confusion, etc. I think on a certain level head-fiers desire to reclaim this, to recreate this pseudo-magical experience, and so they constantly chase one new product after another in an attempt to fill that void they feel, however as soon as they reach a new product---be it a new flagship IEM or miracle DAC---they discover that it's not quite "it" and so they move on to the next one. Desire is only desire insofar as it remains unfulfilled. Audiophile culture largely creates that desire with the promise of a headphone (or any other device) that truly captures the experience one longs for, truly gives the user that unbound primordial enjoyment and sense of satisfaction. Of course such a product doesn't exist, nor will it. But that's not the point. I think we know this on some level, yet we go along with it anyway, placing value in "the journey" so as to maintain our sense of order and purpose within this hobby. It's not necessarily a bad thing in that sense.
However I think this is perhaps the biggest obstacle for some people in having just one rig.
