Do Chinese characters make it a lot harder to create or play word games in Chinese than it is in English? I have seen a few Chinese crosswords, but it seems like they aren't as densely cross-linked as English ones usually are. (In the few that I've seen, Chinese ones seem to have more black squares.) I suspect this difficulty is why logic puzzles like Sudoku and Masyu that use symbols instead of words are popular in Japan.
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.
There is a game very similar to Shiritori in Japanese. It's called "文字接龙" And for more advanced Chinese speakers, all the words need to be idioms. It's called "成语接龙".
Above answers provide a few Chinese word games, but I think most of them are a little too hard for beginners. Here I suggest an easier word game I played with my wife.
At the beginning of the game, you thought (usually) two letters, for example "SC". Then all players must say Chinese words whose “声母” meets SC.
Player A: 市场 Shi Chang -> SC
Player B: 生存 Sheng Cun -> SC
Player C: 时常 Shi Chang -> SC
Player A: ...
During the game the players can learn both PinYin and words.
During Chinese New Year there is 猜灯谜 also called 射虎台。The idea is to deduce people, places, etc from phrases.
Here's a 青少年学生-level example from this web page:
The phrase is 燕姿顿觉终日闲。The answer is 孙悟空。
“燕姿” is 孙, because "孙燕姿", Stephanie Sun, a Singaporean singer well-known in east Asia's Chinese language pop music scene.
“顿觉” is 悟。Self-evident once pointed out. :-)
”终日闲” is 空。 Ditto.
Can be quite tough. I can't do most of these myself. :-)
On the general topic, I recommend heartily this book 中华趣味语文:
(No affiliation. It's the first link from DDG.)
Of course there are many Chinese word games. They just have different style than the English ones since the two languages have different writing systems.
Many Chinese word games are about radicals forming a character, not letters (characters) forming a word.
I did a quick search and found an example: http://www.5dhz.com/game/chengyu/
There are many other games, but the ones I can think of all require two or more people to play, such as "成语接龙" (as other answers have mentioned)，riddles，matching couplets (对对子), etc.
Other than 成语接龙，I've played another word game of 成语. First，given a special topic, e.g. 数字(number)、生肖(Chinese Zodiac) etc， then name one 成语 related to the topic alternately. If you cannot think of a new one in your turn, you lose.
E.g. the topic is 生肖, I start with 鼠目寸光, then you say 对牛弹琴, then a third person would go with 如虎添翼, and so on. If one couldn't name a new one, then he/she lose. Note that you could add some limits to the game to make it more difficult, in this case, a limit is that 生肖 should be in the order of 鼠牛虎兔龙蛇马羊猴鸡狗猪, which means that the next person should say 成语 with character 兔. Actually, there are so many 成语 related to 生肖 that it is not quite hard in the beginning.
In recent months there has been a CCTV game show called 中国成语大会, where contestants play a game which could be called "chengyu charades": contestants form pairs where one must provide spoken hints without saying any character in the chengyu, and the second must guess the chengyu.
This show had a precursor: 中国汉字听写大会 or Chinese Characters Dictation Competition, which is very similar to spelling bees: contestants hear a difficult word and must write the correct characters down.
Both shows contain very advanced Chinese, but if you stick with an easier vocabulary, these games can be very useful teaching tools. You can find these shows on youtube too.
Even though word games are not very popular as before， but you still can find many of them：
And a hard one， 飞花令.
飞花 is a quote from 春城无处不飞花， 令 means 酒令， you have to drink when you fail in the original rule。
The rules are simple， given a character， everyone say a sentence of a poem than include the character at a position。
If the character is 月。
1st player： 月出惊山鸟
It is really hard, I can't play it.
Luckily， there are apps that reduce the difficulty.
You can find these games in Apple app store or Google play, others may in the TapTap. If you dig in more, there are more games, like 猜歌词， 猜电影， 你画我猜...
Those games need you familiar with Chinese pop culture.
What I like is 你画我猜 ↓
Its a Mandarin version of Word Trivia! Its fun and exciting, very challenging and its free!