The incipient infinite cluster is a critical percolation cluster that is conditioned on being infinitely large. Even though the condition has probability zero, Kesten demonstrated the first such construction for critical two-dimensional percolation in 1986 through a suitable limiting scheme. Similar constructions for critical high-dimensional percolation have been carried out by van der Hofstad and Jaraí in 2004. In this talk, I will present a construction of an incipient cluster with *two* disjoint connections to infinity, which we coin bi-infinite incipient cluster. Based on joint work with Manuel Cabezas, Alexander Fribergh, and Antal Jaraí.