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.
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.
Gap risk contracts.
These are daily-restriking putspreads that pay & cancel only if the underlying drops more than (say) 20% as measured vs yesterday's closing level.
Contracts can range from as short as 6 months to 10 years.
Cannot replicate that using Europeans.