It depends on what you consider to be a drinking game, really; we know that poetry and bragging were popular pass-times while drinking during the saga age - Egils saga has a particularly memorable scene where young Egill crashed one such party, and another where he gets overly drunk during such an event and vomits all over his host.
We have other examples of similar activities even in the Eddas, particularly in Þórr ok Útgarða-Loki, where Þórr gets himself into drinking contest that results in him draining part of the seas.
Drinking games were definitely a big part of culture in China. As some mentioned, some were poetry games (go around in a circle where everyone says a line in a poem, whoever can't think of a line drinks), dice or number games, etc. One simple game was to hold up a certain number of fingers on your hands, hiding one hand behind your back and showing the other to the other players, who must guess the total number of fingers you are holding (if they guess right, you drink, otherwise they do!). We definitely have clear evidence from the Tang Dynasty of drinking games, I've heard it dates back to the Western Zhou evolving from rituals, but I'll be honest it's very hard to find sources for this on the internet (most Chinese documents are not digitized). According to Charles Benn in China's Golden Age: Everyday Life in the Tang Dynasty, many drinking games at banquets were very formalized, even including referees, master of the horn, and so on.
Unique to China would be some of the word games that can only be done in their language. A good example is in Strange Stories from a Scholar's Studio, which is a collection of folk tales compiled in the late 17th century and published in the mid 18th century. For example, see here, linking the source, but a 'partially translated' version will make more sense:
Several goblets of wine have been drunken, then someone started a game by using a Chinese character: “The Chinese character ‘ 田 ’is all-sidedly enclosed with a Chinese character ‘ 十 ’in the heart; push the Chinese character ‘ 十 ’upwards, then another Chinese character of ‘ 古 ’ is formed, so I can be excused from a goblet of wine.” A second person picked up: “The Chinese character ‘ 回 ’is all-sidedly enclosed with a Chinese character ‘ 口 ’in the heart; push the Chinese character ‘ 口 ’upwards, then another Chinese character of ‘ 吕 ’is formed, so I can be excused from a goblet of wine.” Then a third person continued ： “The Chinese character ‘ 囹 ’ is all-sidedly enclosed with a Chinese character ‘ 令 ’in the heart; push the Chinese character ‘ 令 ’upwards, then another Chinese character of ‘ 含 ’is formed, so I can be excused from a goblet of wine.” Then a fourth person continued the game, “The Chinese character ‘ 困 ’ is all-sidedly enclosed with a Chinese character ‘ 木 ’in the heart; push the Chinese character ‘ 木 ’upwards, then another Chinese character of ‘ 杏 ’is formed, so I can be excused from a goblet of wine.”Finally, it was Zhan’s turn to carry on the game, but he failed. The others laughed, “Now that yo u have failed, you should drink a goblet of wine.”A goblet of wine was passed to him, when Zhan continued in a hurry, “Alas! I’ve got it. The Chinese character ‘ 曰 ’ is all-sidedly enclosed with a Chinese character ‘ 一 ’in the heart ， ...”The others laughed again, challengling, “Then what is formed?”Zhan had no choice but to drink the goblet of wine, laughing, “Push the Chinese character ‘ 一 ’ upwards, then no Chinese character is formed , so I have to drink up this goblet of wine!”
translation from here
/u/texpeare mentioned one from Ancient Greece in this post