First the typo: 終わりまで -> 終わるまで.
Next you should really consider the meaning of the sentence you are trying to construct as well as the grammar. What do you think もう would mean if the sentence was ゲームがもう終わるまで...? Until the game finishes more?/now?/soon?/already? None of these make much sense to me.
There is another verb in the sentence, and there is another adverb, and it's important to remember that adverbs can modify other adverbs as well as nouns. We have まって meaning 'wait'. This is modified by the adverb ちょっと to give ちょっとまって "Wait a bit". We can then modify all of this with the adverb もう to give もうちょっとまって "wait a bit more/longer".
Altogether we have:
Wait a bit longer until this game finishes.