Say I have infinite precision of strikes $K$ (continuous world $dk$) and expirations $T$ (continuous $dT$) all with liquidity (so no practical limitations). What positions in an underlying can't be replicated? I'm under the impression I can replicate any European payoff, so if I had infinite expirations I could replicate really any position.
Answer
Any non path dependent European type payoff $f(S_T)$ can be replicated in a model independent way with vanilla calls and puts provided $f$ is twice differentiable (in the distribution sense). This is a consequence of the Carr-Madan formula.
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