This game assumes the computations run to completion on systems that will never run out of resources. No one in this universe will ever compute Ackerman's Number, BB(6), or the final answer given in the post. Computations that never complete are infinite.
If you are playing this game and can't produce a number that doesn't fit in this universe you are probably better suited playing something else. That's just table stakes. If it even qualifies as that. "Inscribe every subatomic particle in the universe with a 9 every planck instant of the universe until the heat death of the universe" doesn't even get off the starting line in games like this.
Another general comment: It feels like a lot of people are really flailing around here, and need to understand this is a game. It has rules. If you change the rules, you are playing a different game. There is nothing wrong with playing a different game. It is just a different game. The game is not immutably written in the structure of the universe, or a mathematical truth, it is a human choice. And there isn't necessarily a "why" to the rules any more than there's a "why" to why the bishop moves as it does in chess. You can, in fact, change that rule. There are thousands of such variants. It's just that you're playing a different game than chess at that point. If you don't want to play the author's game, then that's fine, but it doesn't change the game itself. And proposing different solutions is equivalent to saying that you can win a chess game by just flipping over the board and yelling "I win". You can do that. Perhaps you've even won some game. But whatever game you just won, it isn't chess.