On this page, an enlarged variant of Chess is proposed that attempts to add quite a bit to the complexity of the game for the amount that it enlarges the board.

The board is enlarged to ten ranks and twelve files.
Each player has three rows of pieces. The third row from the back contains only four pieces, which, for want of a more appropriate name, I will simply call Checkers, although their move is not quite that of a piece in that game.
Their arrangement, as seen in the diagram, is
-- Ch -- -- Ch -- -- Ch -- -- Ch --
where an empty square, whether white or black, is represented by two dashes.
It may be noted that even with the limitations noted above, the compulsory nature of capturing a Checker provides that piece with an immense power, since it can break up Pawn formations.
The second rank from the back contains twelve Pawns. Despite the enlarged board, on their first move they only have the privilege of moving two squares forwards, subject to en passant capture, not three. In this game, a Pawn may be taken en passant by one of the opponent's Checkers as well as by one of the opponent's Pawns; the Checker would jump over the intervening square the Pawn passed through on its two-square move, to an empty space beyond, and capture the Pawn that way.
The back rank contains the array:
Rook, Knight, Bishop, Tiger, Queen, Man, Leo, King, Tiger, Bishop, Knight, Rook.
Games are scored by means of Dynamic Scoring, allowing a partial win by stalemate, bare king, or perpetual check.
Note that the extensions of either Spectral Realm Chess or Antimatter Universe Chess can be combined with it as they can to ordinary Chess.
In the case of the Checker, there are some important considerations which apply to its move. In the case of Spectral Realm Chess, a Checker can move a half-step diagonally to enter the alternate board, or make a jump of two diagonal half-steps to move one square diagonally on one board, capturing the intervening piece on the other board. In the case of Antimatter Universe Chess, a Checker also can move one square diagonally on one board, capturing the piece on the intervening square on the other board, but this is now a move, rather than a jump, and so multiple steps of this type in a single move are not possible.
Note also that, in Spectral Realm Chess, a series of multiple jumps must all be made in either the normal board of squares, the normal board of points, or the compound board. However, a series of jumps may mix orthogonal and diagonal jumps, and an orthogonal jump on either normal board is also a diagonal jump on the compound board, therefore, orthogonal jumps on one normal board may be mixed with either diagonal half-step jumps, which are orthogonal jumps on the compound board, or with diagonal jumps on that normal board, but not both.
Initially, it struck me that a number of other pieces could be placed where the Man is placed in the array above. Any piece of limited mobility, such as a Fers, a Wazir, a Walker (Wazir-mover, Fers-capturer), a Count, or a Duke would work in about the same way. Or a somewhat more powerful piece, such as a Princess, or even a much more powerful piece, such as the Gryffin, could be used.
For something that would add a more significant novelty to the game, as the Checkers, Tigers, and Cannon do, I have now thought of a suitable piece, which I will call the Flag.
A Flag can be captured like any ordinary piece, but it cannot capture or be moved on its own.
The King, however, on any move where it is not in check, can exchange places with the Flag; this is a normal move, and can happen as many times as desired during the course of the game. Note that the Flag also must not be en prise when this move is made, as the King cannot move into check.
When the King exchanges places with the Flag for the first time, if one or both of the Rooks has not been moved, it regains the power of Castling. Thus, a possible sequence of moves, with the rest of the back rank cleared, would be for the King to Castle Kingside, then exchange places with the Flag, and then, from the Flag's starting square, Castle to the Queenside, and then exchange places with the Flag to return to the snug safety of a Kinside Castled position.
Incidentally, note that the increased defensive resources offered with this variant, which I will call Flag Chess, might be nicely balanced by the increased resources available to the player who has obtained an advantage, through re-entering captured pieces, in Antimatter Universe Chess, making Antimatter Universe Flag Chess a variant of particular interest.
Here, then, is an image of the initial array for the chess of the future, Antimatter Universe Flag Chess:

In addition to balancing the offence and the defence, the branching factor for each move for that variant would be increased, helping to make the game harder for computers. In addition to the large number of possibilities for drops, the ability of Checkers to carry out multiple jumps also provides an increase in the number of possible moves. Also, the positional factor involved with Tiger captures, and the decision to pursue a victory smaller than checkmate, will present challenges to computer players.
Here is a variant played on an 11 by 11 board, which was at one time a popular size for Chess variants with a military theme.

In the front row, each player has six checkers, moving as described above, and five spaceships, as follows:
Ch Sp Ch Sp Ch Sp Ch Sp Ch Sp Ch
Each spaceship has a definite orientation, which can be in any of the eight orthogonal or diagonal directions. Its orientation determines its possible moves, as shown below:

A spaceship moves in the direction in which it is pointed up to three squares orthogonally, and up to two squares diagonally. It may also be rotated by 45 degrees, and then move, but in that case it moves one fewer space in the new direction.
When capturing, as befits a spaceship, it is considered to blast the piece which it is capturing with its ray guns. As a result, the rule for capturing is as follows: a piece may be captured if it is on a square to which the spaceship can move, but the spaceship is to move towards that square, but stop one square short of it.
A spaceship can therefore capture a piece on an adjacent square in either the direction in which it is pointing, or one of the directions 45 degrees on either side of that direction, without being moved, but with being rotated, if necessary to have it point towards the captured piece.
There is one additional rule, to deal with the case of spaceships that reach the edge of the board and cannot move. It is possible to turn a spaceship 45 degrees, without moving it, and without making a capture, if after that move is made, the player who does so also then moves another piece which is not a spaceship. This deals with the issue without promoting spaceships, giving them a backwards move, or eliminating the possibility of Zugzwang from the game.
The second rank is a line of pawns.
On the back rank, the pieces are:
Rook, Tiger, Camel, Knight, Bishop, Queen, King, Bishop, Knight, Camel, Rook
The Tiger moves without capturing like a Bishop, and makes captures only like a Knight. Note that the Tiger is on a white square, so it initially only attacks black squares, which are also the squares on which all the checkers are located.
The Camel, which makes a leaping move similar to that of the Knight, but with a displacement of three spaces in one orthogonal direction, and one space in a perpendicular direction, thus remaining on the same color of square, is the remaining unusual piece in this game.
An alternative form of this game may be played with the Queen's Rook replaced by a Cannon. The Cannon moves like a Rook, but when capturing, it captures the second piece on any orthogonal line, jumping over the first piece. In this case, Queen's side Castling still takes place normally, with the Cannon acting as if it were a rook.
Here's another alternate form of Chess, only slightly expanded, on a 12 by 8 board, and yet I think of some interest:

This variant is intended to occupy a place halfway between the old form of Chess, in which the Queen had not replaced the Fers, and the Bishop had not replaced the Dabbaba, and present-day Chess.
The pieces along the back rank are:
Rook, Cannon, Camel, Knight, Bishop, Tiger, King, Bishop, Knight, Camel, Cavalier, Rook
The powerful Queen is replaced by the less powerful Tiger, which moves, but does not capture, like a Bishop, and which captures, but does not move without capturing, as a Knight.
The Camel is, as we've seen, a piece that moves like the Knight, but the long part of its move is one square longer, resulting in a displacement of one square in one orthogonal direction, and three squares in a perpendicular orthogonal direction.
The Cavalier, depicted in the diagram by a checker, moves one square orthogonally or diagonally, but does not capture in this fashion. It may jump over an allied man to move two squares without disturbing that piece. As well, it may jump over an enemy piece, capturing it. It may also, in one move, make multiple jumps, all of the same kind (that is, all jumps over allied pieces, or all captures of enemy pieces), of either of these two kinds. As well, it may make one or more jumps over allied pieces strictly followed by one or more jumps over enemy pieces. (This move may be familiar as the "Knight's Charge" from the games of Chivalry and Camelot by Parker Brothers.) In either of these cases, however, either all of the jumps made during a single move must be diagonal, or all those jumps must be orthogonal. This is the one difference between the movement of this piece and that of the Checker in the first variant on this page.
The Cannon in this game is actually a Vao. It moves like a Bishop, and captures the way the Cannon does in Chinese Chess, by jumping over one intervening piece (the "screen"), but again along the diagonal lines along which a Bishop moves, rather than the orthogonal lines along which a Rook moves.
The Rook also has a changed move: it still moves orthogonally, but it can only make moves from one to five squares in length.
Castling is as in regular Chess; the King moves two squares, and the Rook, provided no pieces are intervening, moves to the space the King passed over.
The Pawn may move two spaces forward on its first move, and, as well, en passant capture is the same as in regular Chess.
As well, the places of the Cannon and the Cavalier may be interchanged once, immediately before the first move of either of those pieces (the other of which must also have not been moved). The purpose of this special move is because the King's side Cavalier can move without disturbing the wall of Pawns in front of what could become a Kingside castled position; thus, it facilitates Queenside - or, rather, Tiger side - castling.
Thus, a single move may, when neither a player's Cannon nor the same player's Cavalier, has been moved, consist of first interchanging that Cannon and Cavalier, and then moving one of them for the first time.
Also, if the Cannon and Cavalier are interchanged before the first move of either, that first move may not be a capture. One consequence of this restriction is that the Cannon only pins the opposing player's King's Bishop Pawn at the start of the game, and not the Tiger's Bishop Pawn as well.
Thus, as the Queen has been replaced by a piece with roughly the power of a minor piece, and the Rook has been limited in range, it is presumed to be less likely that a game will be decided in the first few moves, thus, it is hoped, reducing the tendency to stereotyped openings.
The Tiger changes color with every capture, and the Cannon and even the Cavalier have sufficiently unusual moves to add interest and surprises to the game.