You can try to see your purchase history on the web. This will include all purchases you have made as well as apps and games that you have installed on the various Apple devices you have used. This will be identified using your Apple ID.
See this:
How to see recent purchases on the web
- Go to reportaproblem.apple.com.
- Sign in with your Apple ID and password.
- A list of your recent purchases appears. If you’re not sure what you were charged for but you know the exact amount, search for the amount. If there's a problem with an item that you purchased, use this website to report the problem to Apple.
To see charges from older purchases, you might need to look at your purchase history in Settings or the App Store.
Learn what to do if you don’t see the item you’re looking for or if you see unexpected charges.
Using the above steps, you might recognize the game app you are looking for.
Good Luck!