The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite.
The function SSCG(k)[1] denotes that length for simple subcubic graphs.
The function SCG(k)[2] denotes that length for (general) subcubic graphs.
Adam P. Goucher claims there is no qualitative difference between the asymptotic growth rates of SSCG and SCG.
"[3] The function was proposed and studied by Harvey Friedman.