We introduce the largest connected subgraph game played on an undirected graph $G$. In each round, Alice colours an uncoloured vertex of $G$ red, and then, Bob colours an uncoloured vertex blue, with no vertices initially coloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph of order greater than the order of any blue (red, resp.) connected subgraph. If neither player wins, it is a draw. We first prove that Bob can never win, and define a large class of graphs ( reflection graphs) in which the game is a draw. We show that determining the outcome of the game is PSPACE-complete in bipartite graphs of small diameter, 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 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.