Bernhard Keller introduced maximal green sequences as a combinatorial tool for computing refined Donaldson-Thomas invariants in the framework of cluster algebras. Maximal green sequences furthermore can be used to prove the existence of nice bases of cluster algebras and play a prominent role in the work on the full Fock-Goncharov conjecture due to Gross-Hacking-Keel-Kontsevich. In Physics, maximal green sequences appear in the computation of spectra of BPS states. We report on joint work with Gleb Koshevoy introducing maximal green sequences for certain triangle products of quivers. As an application we comment on the consequences regarding the full Fock-Goncharov conjecture for double Bruhat cells.