shiritori is a fun game to play you can start to play once you have a good amount of nouns to your knowledge. (you are not allowed to use verbs nor, I think adjectives (maybe just i adj. but as with most games, I’m sure it depends who you play with))

henohenomoheji isn’t so much of a word game, but it is interesting. One of my students actually taught me that just a few months ago.

but my favorite is dajare. (oyaji-gyagu). As someone learning the language, making pronunciation mistakes or confusing similar sounding words was very common. So it was common for me to pull them together to try to make light of my mistake and get a laugh out of people. Its definitely fun to play with words like this!