A long comment, too long and painstakingly difficult to keep re-editing in the comment boxes:

If you don't **require** a draw to be declared, there are multiple scenarios in which king vs. king or (king+queen) vs. (king+queen) can play on infinitely; in that case, the game tree of chess is unbounded. There must be a strict rule for when to prune a branch in the game tree. @Didier-Piau, the upper-bound concept as posited by the poster of this question appears to have 3 mistakes in it.

It may be the concept of {white pawn, white other, black pawn, black other, empty}$^{64}$, which has a set size of $5^{64}$.

This makes the mistake of lumping all of the pieces into $4$ categories. Even if you define the pieces to be {Black, White} $\times$ {Pawn, Queen, King, Rook, Knight, Bishop}, and allow for an empty space, then $13^{64}$ would be a better (but still grossly overlarge) upper-bound on the number of chess board configurations as it included multiple implausible configurations with an impossible count of pieces. A better guess might be the combinatorial (64 choose 32) + (64 choose 31) + ... (64 choose 1), and that can be pruned in many ways such as if the last board position has only one piece in it, then that last piece could only be the winning side's king, etc.

It makes the mistake of conflating the number of possible positions or "boards" of a chess game with the number of paths through these possible boards; this is equivalent to the error of confusing the number of vertices in a directed graph with the number of paths leading out from a starting vertex.

And it makes the error of not being rigorously defined: for example, defining the tree correctly, as the tree starts out from a fixed board position.