Nope, games without any genre would fall into the "games with no genre" genre.

People will either make up a new label for a game, label it 'experimental' or 'indie' if one can't be easily thought up or shoehorn it into a combination of other labels. Why? Because for people to describe how great or horrible a game is they need to reference something, and doing that will put a label on it. The exact same thing with trying to market a game, the people doing the marketing need to know what demographic to market the game towards.

Everyone feels an overwhelming need to classify things as type "x" or "y" but you can't always fit everything into a cute little box all the time! Flower is a great example of something you can't quite classify but people still try. It's easier to tell your friends to try a game when you can say it's like a cross between Street Fighter and Metal Gear then "Oh just try it dude you'll like it". I personally think we need to make more games that you can't easily fit in a round hole! I mean when GTA III first came out no one really knew how to classify it and now look at how many GTA clones/open world games there are out there it's become an entire genre on to itself.

You can make a game where their are many features or references to different game genres but this is a question of classification. Classification is a thing held to organize products when that thing is in a group of high numbers. So things will always be in some kind of genre class. Games like flower or linger in the shadows are considered puzzle games, where as ICO is considered an adventurer game, even with its heavy emphasis on puzzles.

Everything has a genre... it's how you find games that might be of interest. Even "other" would be a genre... ne?

I don't think so. Mostly because the instant something new is created it will immediately be placed into a new genre. So while I believe it's possible to generate new genres, I don't think that you can create something entirely out of genre conventions.

Rez is a conceptual rhythm game. Flower is a "visual poem". Heavy Rain is a Noir immersion experience.

I mean, genre is such a broad term. But I don't think it's at all limiting. It's just a social construct. New and interesting films are constantly being produced, all of them with very clear genre keywords. But products of quality can be made regardless. (This is arguable I guess, but I like to think so.)

So while I don't think you can generate a game without it falling into a genre, I also don't see this as a bad thing.

Yes.
But it is difficult.
Firstly, due to publishing and money. Finding someone willing to fund your dev house requires a design doc that sounds profitable. no one wants to hear, "I have this idea for a game, no one has ever thought of before", they want to hear "I have this game that is like Zelda but better, so will sell well, and have sequel potential".

Secondly, due to large numbers of genres. Merging genres is pretty popular now. FPS's with rpg elements, RTS, with FPS elements, and so on. Simply put, any attempt to make a game, will very likely step into the realm of one of the existing genres at least a little bit. You would almost specifically have to be trying to avoid any to not be one.
Which then asks the question, are you trying to be different, or make a fun game? And making a game fun, is more important then being differant for the sake of being differant.

Make a game about Cthulhu... it is undescrible and our mind cannot grasp its epic story or features.... huhuahuahuahauhau

i don't think game companies would have a good enough time trying to market/attract the desired audience WITHOUT putting it in a genre

Yes! Its called watch paint dry story! all you have to do is press a button and watch paint dry! you can't really call it fun but hey a games a game XD