Oh, I remember the convection cell Qax from Baxter, read
Dragon's Egg when it came out, or Brin's
Sundiver for solar dwellers, or go even further back for the Piers Anthony's
OX, even the frozen lattices of light from
2001. I could try speculating on some pretty strange forms of life, and have, but ultimately life -- and as fuzzy as that term is -- might well be construed as interlocking sets of cycles, self-perpetuating at the expense of various resources, always with the second law coming up from behind, waiting to knock it all out of balance, pluck teeth from the gears.
I could talk about how the various conditions at the surface of a neutron star probably cannot support the kind of interactions that would lead to a nuclear pseudo-chemistry (the elements aren't as important so much as the kind of bonds, and therefore structure supported, are, with the other physical properties being less important), but I would like to take a different tack and think about life arising. How would all of those little cycles, circadian down to Krebs, get started?
My intuition? guess? is that just as we are big bags of internalized seawater, life would have hijacked, replicated, and then encapsulated already existing cycles, multiple cycles. Just once cycle wouldn't cut it, probably not even two. I think for us it was all about warm little tidepools, water comes in, then the day-night cycle keeps changing the concentrations. Another tide comes, more water and something, perhaps waste, is washed out. Enough times, enough luck, maybe something sticks, holds something back in case the next splash isn't as big. The next big trick is a membrane, something to isolate the cycles. Separates the outside from the inside. Now there's something to separate.
Overall, though? You're still going to need matter that supports both structure (at least for a membrane, a boundary between in and out) and fluidity (for those cycles). I've thought a lot about what the bare minimum would be, but the Earth chemistry is just a shorthand for "what bonds, and therefore structure, can be maintained?" and "how will it flow?" I am pretty confident in saying it would have to obey Fermi statistics, instead of Bose, because bosons love to play follow the leader, but fermions need to stand out (and apart) enough that they can at least stand. The convection cell entities were interesting to think about but would require an almost unparalleled level of stability and uniformity across large expanses of matter, sort of a large-scale "lab conditions" environment, free of fluctuation and interference, somewhere relatively calm. I have considered something not too far away from that, life existing in an environment where the average energy levels were constant, but the entropy wasn't: life optimized to hunt not for calories but for pockets of order to bolster its own.
Non-chemical life is fun to speculate about but ... I keep coming back to structure and fluidity. It's difficult to find phases of matter that support both that aren't in the chemical domain.