In each round of the largest connected subgraph game, Alice first colours an uncoloured vertex red, and then, Bob colours an uncoloured vertex blue, with all vertices initially un-coloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if the connected subgraphof maximum order is red (blue, resp.). We first prove that Bob can never win, and define a large class of graphs, called reflection graphs, in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognizing reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or of sufficient length, and that Alice wins in cycles if and if only if the cycle is of odd length. Lastly, we give an algorithm that determines the outcome of the game in cographs in linear time.