Since you are being very specific about the game of Go, I will be very specific in my answer.
If talking about the game, you would not pluralize Go at all.
Comparing it to chess (or any other game):
✔ How many games of (Go / chess) did you play?
✘ How many (Gos / Goes / chesses) did you play?
✔ I'd like to play three more rounds of (Go / chess).
✘ I'd like to play three more (Gos / Goes / chesses).
Whether it's Go or something else, the names of games keep the same form. It's games, rounds, or some other descriptive word that gets pluralized instead.
Note that some names of games are pluralized in the first place—such as darts. You would play one round of darts or two rounds of darts (not dart).
So, in the domain of language related to games, the base form of the name of the game stays the same, whether the name uses a singular or plural form.
Update: As asked in a comment, I will try to add some kind of authoritative source for this.
I suppose it's possible that in conversation Go could be pluralized. Many ungrammatical things are done in conversation that would never be done in writing. It's even possible that examples of it could be found in informal writing. But unless there is a preponderance of such usage, I would say it is at best obscure and at worst simply aberrant. (Although useful as a communication tool among those doing so.)
Intellectual ownership is one explicit argument against the pluralization of a trademarked proper noun.
Chris Scott Bar discusses this in the article "The plural of iPad is ‘iPad devices’ says Apple (LEGO, too!)":
Yesterday I Photoshopped a picture of a stand that I built for my two iPads out of LEGOs. Do you know what’s wrong with the previous sentence (except for the fact that I did no such thing yesterday)? I used all three of those brand names incorrectly, according to the companies that make them.
You might already know about using Photoshop or LEGOs incorrectly, but iPads? That’s a new one to me. Yesterday, Apple’s own Philip Schiller took to Twitter to settle a dispute that can cause a bit of confusion. Namely, what is the plural of iPad Pro? Is it iPads Pro? iPad Pros? As it turns out, the correct way is “iPad Pro devices.”
Mr. Schiller pointed out that you should never pluralize any Apple product. This goes for the iPad, iPhone, and even a Mac. That’s right, if you have more than one Mac, you should say “I have 3 Macintosh” or “I have 3 Macintosh computers.” Yes folks, Macintosh, much like deer or fish, is plural. Of course, no one has actually said the word Macintosh since the 90s, so it’s a moot point.
Apple isn’t the only company to get weird about how their names are used. Plenty of other companies have gotten huffy at the thought of their names being used improperly. Many dislike their names being used to describe any similar product, such as Kleenex for a facial tissue, and Band-Aid for an adhesive bandage. But here are a few fun ones that you might not know.
Photoshop is a verb that gets used very frequently. Just the millions of people on the PhotoshopBattles subreddit. You may not know it, but Adobe doesn’t take kindly to that usage of their name. In fact, they have an entire page on their website dedicated to their trademark guidelines. They don’t want you to use it as a verb, a noun, or in the possessive form. Remember, instead of saying “The image was Photoshopped” you should say “The image was enhanced using Adobe® Photoshop® software.”
Since I was a child, I loved playing with my LEGOs. I’ve always said it that way, and I pretty much still do. However, LEGO doesn’t appreciate this. For years, if you tried to go to LEGOs.com you would receive a special message from the company letting you know that this is an improper use of their brand name. They remind us that the correct plural form is LEGO bricks, or LEGO toys.
Google is another interesting case, as Merriam-Webster has added an entry for the word “google” in their dictionary. Their entry still specifically states that it means “to use the Google search engine to obtain information about (as a person) on the World Wide Web.” The company actually made a blog post back in 2006 to address the concerns about their brand name being used improperly. Essentially, they want to make sure that you’re very clear that when you Google someone, you’re actually using their services.
This is the specific legal text from Apple on never capitalizing the names of its products:
Rules for Proper Use of Apple Trademarks
1. Trademarks are adjectives used to modify nouns; the noun is the generic name of a product or service.
2. As adjectives, trademarks may not be used in the plural or possessive form.
Correct: I bought two Macintosh computers.
Not Correct: I bought two Macintoshes.
Or in terms of the legal guidance from LEGO:
Proper Use of the LEGO Trademark on a Web Site
If the LEGO trademark is used at all, it should always be used as an adjective, not as a noun. For example, say "MODELS BUILT OF LEGO BRICKS". Never say "MODELS BUILT OF LEGOs".Also, the trademark should appear in the same typeface as the surrounding text and should not be isolated or set apart from the surrounding text. In other words, the trademarks should not be emphasized or highlighted. Finally, the LEGO trademark should always appear with a ® symbol each time it is used.
Discussing the specific trademarks used by the games of Go and chess likely makes no sense, because they exist in the common domain and have no trademark holders to enforce any kind of naming. But there are trademark owners of other games who might challenge the pluralization of the names of their games in a fashion similar to Apple and LEGO.
Still, in the domain of the language of games, Go and chess fit in with other such commercial games.
There is no authoritative source I could find that says Go shouldn't be pluralized—but there is also no authoritative source I could find that says it should be pluralized. Can it be pluralized? Certainly. Should it be pluralized? That's a matter of opinion.
So-called grammar rules aside, it's purely a matter of personal style and choice. People can use words as they wish. If the people they're communicating with understand that usage then those words have served their function.
Still, the pluralization of the names of games seems to be something that's mostly done conversationally and seldom in formal writing. Pluralizing the noun Go rather than using other words around it to form a plural is certainly the least common of usage choices. (At least when it comes to writing and filterable game names, this can be shown through various Google Books NGgram Viewer queries.)