Free rieg elements as games

Question

Let $F_1$ be the free rieg on one generator $x$. Can its elements be interpreted as combinatorial games? Can the unique rieg homomorphism

be interpreted in terms of those games?

What an answer must provide

Specify the games and their equality, interpret $0,1,+,\times$, and exponentiation, and show that the rieg equations hold. A full model of $F_1$ must also identify when two game expressions represent the same element; a map out of $F_1$ alone need not be injective.

Next step

Give a concrete candidate for exponentiation of games, then test the exponent laws and the evaluation $x\mapsto0$. The existing recursive-coalgebra model of impartial games does not by itself supply these rieg operations.

Sources: original problem 7.0.1; Classical vs quantum computation.