It depends on how and where you log in your Facebook account. You might not see the "x" button if you're using mobile phone unless you will use any search engines like Google Chrome or Internet Explorer and log in to your Facebook account in a desktop mode. There you will see the "x" button to delete game requests. However, the way I understand the question by @Susan, she doesn't want to receive any game requests by anyone and I do have the same issue. What we (for sure) both like to happen exactly is to prevent anyone from sending us games requests and we don't want to delete and block each game request one by one or individually. In short, we want to automatically block/cancel games request.
I think that most of the answers here are correct and helpful and let me just clarify and simplify the answers as wee really cannot automatically block all games request so we wont see them anymore.
Answer # 1
1. Click the "Setting" button. (usually looks like a snowflakes or a gear on the top)
2. In the left side, look for "Blocking" button and click it.
3. It will then show you "Manage Blocking" page that has 5 sections listed below.
(Restricted List, Block users, Block apps invites, Block events invitations, Block apps)
(Focus only with "Block apps invites" and "Block apps")
4. In the "Block apps invites" section, type the name of your friend who always send you games requests then hit enter and Facebook will prevent that person sending you games request.
(You're still friends with that person as you only prevent him/her from sending you request)
5. In the "Block apps" section, type the name of the App you want to be blocked so no one could send you a game request for that particular game.
Answer # 2
1. Go to the homepage by clicking "Home" button found on top of the page.
2. Look for the word "App" found in the left side of your screen.
3. Click "Games" that has a letter F logo on a cube-like box.
(You will then be redirected to a new page that says "Apps Center")
4. In the left side of the "Apps Center" page, scroll down then click "Requests" button.
5. In the middle or center part of the page, all games requests will show up.
6. In each game request, you have 2 options to select/click. "Play Now" and/or "X" buttons.
--> Each game request is boxed/classified by whoever send it to you.
--> Multiple names might show up per game request whoever send you a game request
--> Example: "friend A, friend B and friend c invited you to try Criminal Case.
7. Click the "x" button to cancel/block the game request and it will give you a message and 2 options to select from then click whichever you want to. (See example below)
"You hid 27 requests for Criminal Case"
Block Criminal Case?
Ignore all requests from Friend A?
--> When you block the game, no one will be able to send you a game request for that particular game.
--> When you ignore all request from a particular friend who send it to you, other friend of yours can still send you game request for that particular game and the game is not yet blocked. Also, when you ignore requests from particular friends, you're not blocking your friends.
8. Repeat step number 6 and 7 for the rest of the games.
These 2 answers are very common and these are the only way to solve our problem regarding games requests unless you deactivate your account.