We introduce the largest connected subgraph game played on an undirected graph G. Each round, Alice colours an uncoloured vertex of G red, and then, Bob colours one blue. Once every vertex is coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than that of any blue (red, resp.) connected subgraph. If neither player wins, it is a draw. We prove that Alice can ensure Bob never wins, and define a class of graphs (reflection graphs) in which the game is a draw. We show that the game is PSPACE-complete in bipartite graphs of diameter 5, and that recognising reflection graphs is GI-hard. We prove that the game is a draw in paths if and only if the path has even order or at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd order. We also give an algorithm computing the outcome of the game in cographs in linear time.