Are there word games in Chinese?
Well, not sure if this answers your question or not, but I've played a game before involving using idioms (aka 成语) where you have to carry on using the same sound (not including tone) from whatever 成语 the previous person said.

For example, if I started with 骑虎难下 (qi hu nan xia - something like stuck between a rock and a hard place) the next person might say 下里巴人 (xia li ba ren - like redneck) the following 仁者见仁 (ren zhe jian ren - different people have different opinions) and so on. It's pretty hard, but a great way to learn new 成语.