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The C=C=C=... structure doesn't seem to form, because polymerization usually results in single bonds between the monomer units. But we've observed something similar, namely, polyynes, with alternating single and triple bonds: ...-C≡C-C≡C-C≡C-....

Taking this to infinity, we end up with the hypothetical Linear Acetylenic Carbon (LAC), which has a decent Wikipedia article: https://en.m.wikipedia.org/wiki/Linear_acetylenic_carbon.

Take note, though, that any finite chain is going to have to be "capped" on the ends by something that probably isn't carbon (for example, a hydrogen atom on each end), because the carbon on the end has one remaining bond that has to attach to something. You might be able to bend the chain around into a ring to avoid this, but triple bonds are pretty rigid so this might not be possible.

That said, there's a much more well-known example of carbon-only compounds: buckyballs, made of 60 carbon atoms arranged in a truncated icosahedron.

As just a single chain? No, I think you'll always at least end up with a couple of Hydrogen atoms at the ends.

However, carbon nanotubes can be very long molecules and are comprised of just a single element (carbon). They aren't a single large chain, but rather a hexagonal lattice (graphene) that has been rolled up into a cylinder

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