FYI: The problem of finding the longest "chain" is NP-complete, even in the case of just 2-word phrases, as can be seen from a reduction from the longest path problem in directed graphs.

Each single word is a vertex, and there is an edge from word A to word B if "A B" is a valid 2-word phrase. A path in the graph corresponds to a "chain" of valid 2-word phrases in this game, and we want to find the longest one.