I would write:
I have not heard of this new game called "Why First?."
Do you like the game called "Why First?"?
Wow.
The first example has changed the name of the game to which you are referring. The second example is contrary to your position - you say punctuation should go inside the quotes.
This is disingenuous at best. You are making the argument that, by including the end punctuation of the sentence within the quotation marks, I have somehow changed the actual name of the game, or your understanding of the name of the game. That seems to be many people's argument, in fact -- that the inclusion of a period or a comma within the quotation marks somehow physically/spiritually/genetically modifies the quote itself.
I call bullshit.
Let me try to make up a computermathrobot example. Imagine there's a program or something on a computer that requires the human using the computer to press a series of keys on the keyboard in a specific order, separated by pressing the enter key. Imagine those keystrokes are:
ad,f93'.39*36/;93GF,BY
zR.we;l3-5*&%^a5-lo
meo.412834+)_2c,v23.23c
Now, I've been tasked to write a short instruction for the user that explains that they must input those three strings of keystrokes, with each one followed by pressing the enter key.
I believe that you all have argued against me by saying that the following sentence would be unclear or difficult or something.
To start the program, the user must enter "ad,f93'.39*36/;93GF,BY," "zR.we;l3-5*&%^a5-lo," and "meo.412834+)_2c,v23.23c," following each series by pressing the enter key.
Is that your argument? Is your argument that the reader might accidentally include my commas in their keystrokes, thereby not succeeding at starting the program?
Because that means your argument is that I don't know how to use commas at all to separate a series of items (oxford comma discussion notwithstanding), and that I just wrote a series of items without commas at all. And how is that a logical argument to make at all?
Isn't the logical assumption that the commas are separating the three items? Or do you truly think it is more logical to assume that I do not know the basic use of commas?
If you read that sentence and actually would consider that the commas are part of the string of characters, that's your own fault: you may have assumed I am an uneducated writer who doesn't know what a comma is. Or maybe you are ignorant of the specific rules of the style guide I'm following. There may be many reasons you, the reader, make that mistake. But it isn't because I made an error.
(Note: I'm not arguing that the example sentence is good; I definitely would have written it differently, listing each item on a separate line, etc. I'm just illustrating my point.)