A nim game variant with pass move This variant is almost similar to the normal nim game which states - 


  Two players take turns to remove one or more items from a single, non-empty pile. The player who removes the last item from the last non-empty pile is declared as a winner.


The twist is an inclusion of a "pass" move for each non-empty pile. To make myself clear, a player can choose to pass his move on any non-empty pile on which there was no "pass" move made.

I looked up grundy numbers to solve this problem, but I do not how to add the rule of pass move.

For ex:

$G(0) = mex(empty-set) = 0$

$G(1) = mex(G(1),0) = ?$ 

Heremex(g) is a function which takes a set of non-negative integers and outputs smallest non-negative integer that is not present in the list.

For ex : mex(0,1,2) = 3 and mex(0,2,3) = 1

I just don't know how to proceed. I'd appreciate some help. Also, I'd like to add that the number of piles is small say 10-15 but the number contained in each pile is quite huge. Hence those recursive backtracking type of solutions won't work.

Update : The pass move is not mandatory.
You have to augment each pile with an extra bit which states if it has been passed or not. Then you can compute the Grundy function as usual. For example, here are some values:
$$
\DeclareMathOperator{\mex}{mex}
\newcommand{\yes}{\text{Yes}}
\newcommand{\no}{\text{No}}
G() = \mex(\emptyset) = 0 \\
G(1,\yes) = \mex(G()) = \mex(0) = 1 \\
G(1,\no) = \mex(G(),G(1,\yes)) = \mex(0,1) = 2 \\
G(2,\yes) = \mex(G(),G(1,\yes)) = \mex(0,1) = 2 \\
G(2,\no) = \mex(G(),G(1,\yes),G(2,\yes)) = \mex(0,1,2) = 3
$$
Here Yes means that the pile has been passed, and No that it hasn't been passed.

You mention that a recursive solution won't be efficient enough. In that case, I suggest trying to find a formula for the Grundy function. Here is a hint:


  A pile which has been passed behaves like a normal pile. A pile which has not been passed behaves like a pile which is one item larger.