combinatorial game theory - Do there exist chess positions that require exponentially many moves to reach?

With n/100 black kings, one white king, its possible.

The idea is to make a "switch", a room which is in one of two states, has 1 entrance, e, and 2 exits; x and y, the white king enters the room through a tunnel of pawns, if its in state 1, it must exit through x, while switching the state of the room to 0, in state 0 it must exit through y leaving the room in state 1.

With k switches labeled 1 to k, connect all the x_i with e_1, and y_j to e_(j+1) for all j. Start it with king in e_1=x_i, and all switches at 0, to turn the last switch on youll see that you need to transition thorugh all possible states, giving a minimum 2^k moves.

The switch really needs a diagram, but the idea is to have a black king for each switch, the black king will block a black rook preventing check, so the white king can get to a next room, where it will block a white rook, so that the black king can get to a third room, blocking a black rook, having the white king exit, the black king cant go back without the white king, but it can continue to the black entrance of a second such sequence of rooms, the black exit of which is connected to the black entrance of this sequence. The two white entrances are connected, and the two white exits are x and y. So the black king is in one of two tunnels, corresponding to the states 0/1, which determines which exit is available for the white king.

There might be some way to construct the switches without kings aswell.

There should be a row of pawns below and above the diagram of the switch component.
Couple 4 of these of each color to make 1 switch.