Despite the above, you will never ever, ever in your lifetime see the following consecutive flops: 22x, 22x, 22x, 22x, 22x! It ain't gonna happen.
Really? Why not? Me in particular or any single person in the world? Just in my (or their) lifetime? Or in the life of the game and history of the world? Short of including everything ever, you are making an intentionally narrow window of time for something to occur. That can be done with anything simply by picking a long shot event and narrowing the time sufficiently. I won't be hit by lightining during the 30 second period that follows completion of your reading this whole response.
The
odds of seeing the flops you indicated are exactly the same as seeing the flops 344, K2K, 55J, Q77, AA9 come out in any order. Each flop is an independent event from the previous flop. The only reason you even make this claim is because the grouping of flops you noted shows a significant and easy to remember pattern (which the human mind looks for by nature and is typically wired to see). Opposed to the flops I noted which are not patterns (with the exception of pairing each one to keep probabilities exactly the same as yours).
What's your take on a lottery where you pick 6 numbers from a pool of 55 numbers? Is it possible for the winning number to ever be 1 2 3 4 5 6? Sure. Has it ever been? No, not that I've ever heard. Why? Because that would be such a significant and recognizable pattern that it would have made national news.
So, I'd be willing to state that somewhere between the beginning of flop
poker games and the end of poker, based on all previous hands ever dealt anywhere and all hands that ever will be dealt, any of them online or live, that somewhere among all those cards will be the pattern of flops you seek.