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Hi, George. While I'm not up on my math, I think the numbers of chess variants are limited. Instead of building up from the bottom, taking games like Falcon Chess and showing how each variant can be modified in all sorts of ways, giving zillions of games, let's look at it the other way, from the outside rather than the inside, so to speak. What are the hard limits of chess? There are 3 components to a game, the board, the pieces, and the rules. 
Board size: it can't get much smaller than 1 or 2 squares, nor can a game of chess go much larger than a 100x100 2D board. Even the 'infinite board' variants really don't need more than maybe 1000x1000, and realistically, can be pretty much played out on 100x100 without too much trouble. Let's take 100x100 as our top size, then, for a board; 10,000 squares should be enough room for most games of chess. And we can deal with the rest later, should that be necessary.
Numbers of pieces: If you assume 50% board coverage, then 5000 pieces is about the maximum number you'd want on the board. That seems a bit much for me. Without some tricky movement rules, I've found that 100 pieces per side is a bit much. And without some tricky piece design, 25 different pieces per side is also a bit much. Even if you go with 1000 pieces per side, and 100 different pieces [a team chess game if I ever saw one], that's still a  number we're familiar with. And running all the permutations through just gives you a very big number. And only potential, not actual, games.
Rules: here's where we get crazy. This is where we think all the infinities come in. And we all too often design game systems instead of games, adding to the mess. And what else we often do is mistake the potentialities of the system for the actualities of games. The map is not the territory. Further, while we're into the gadjillions of potential games through permutations, all these games still, in principle, are countable - we're doing it ourselves as we go along. So let's say that the total number of permutations to any starting game is [on the order of] one bazillion, with the specific value of bazillion to be determined [by actual count] in the future. 
Number of chess variants: Currently, there's about 1000 - 10,000 CVs, giving somewhere between 1 and 10,000 bazillion known total potential CVs. Since they're made by people, one way to look at it is that unless there are people and chess forever, the number of variants cannot be infinite. Current theories of the universe favor the less than infinite position.
How many ways can a piece move? On a board of 1 - 10,000 squares, how many different ways can a piece move? ... ... ... Now, dump the really stupid ways. Does that add up to infinite? [Even putting the really stupid ways back in doesn't; heck, people with computers'll play and do darn near anything. And, judging by the spam I get, they think I will, too!] 
How many different boards are there? Well, this has gotta be a very big number, but it has to contain between 1 and 10,000 or so squares that have to be connected in some sorts of ways for the pieces to interact. If you dump the turkeys, it becomes a somewhat manageable number, at least conceptually.
Each new game is going to add either a new bazillion to the total number of CVs, or is going to increase the size of a bazillion for all the other games [at least; truly innovative games may do both, more than once]; in either case, it's countable. 
Chess occupies a limited area in 'game space', that conceptual area where all games are found and [somehow] categorized. And it is a game of discrete parts, digital in nature rather than analog. Pieces and board positions come in units: in chess, the number of squares on the board is set in the rules [even if the rules allow changes, these are determined by the rules] and the number of pieces on the board is there for all to see [and count]. Changes in this number of pieces are again determined by the rules and the actions of the players as allowed by the rules. But all changes are in discrete units, going up or down some whole number, of the limited number of pieces or squares we can have in a game that can be called 'chess'.
Given that humans play these games, either on boards or on computers, I think a case has been made for a very large but countable number of realizeable games. And even playable games. And given the limitations of humans [and you can interpret that broadly to include until-now hypothetical intelligences if you wish], I think it's more reasonable to assume there is a limit to the number of decent chess variants, and, by extension, all chess variants. And while the former number may well be a question of taste, the latter number is easily seen even without calculation to be a number far huger than the total number of chess games ever played or, most likely, to be played. 
So, the number of variants is effectively, if far from actually, infinite, but most of the best pickings are near the top. While many weird and complex games may gain great stature in the future, I think that the percentage of 'hits' will be higher in the smaller and simpler games, and spread out in the larger number of more complex games in a pattern much like that of prime numbers. 
Enjoy,
Joe