In Winning Ways Volume 2 (pg. 399) they state that "since ☽ represents the empty set, we have the obvious addition rules":
$$☽+*n=☽$$
$$n=0,1,2,...$$
I would think this extends to transfinite nimbers, however it isn't explicitly stated.
Is $☽+n=☽$ true for finite & transfinite nimbers?
It isn't explicitly stated because that section of the book does not discuss transfinite/infinite games at all.
That said, I expect the theory of "infinite (nonloopy) games with entailing moves" to work basically the same as the theory for "finite games with entailing moves" presented in Winning Ways, so it's probably still true for transfinite nimbers.