Here is a game for any plural number of players.
(And no grids!)
Players take turns, on each turn writing any positive integer to the end of a growing list of integers. The players do this until there is a predetermined number of integers in the list. (The number of integers is a multiple of the number of players.)
Players then take turns; on each turn a player rewrites the list with one of these changes:
The player changes one integer in the list to a product of any number of PROPER divisors, each divisor greater than 1, that all multiply to that integer. (The divisors must be placed between the same integers in the list as the integer they replaced, but they can be in any order amongst themselves between those integers.)
The player replaces two adjacent integers in the list with their sum. (The sum goes in the same location within the list as the two integers it replaces.)
The first player to get a list completely of primes is the winner.
If a large predetermined number of moves have taken place, and no one has yet won, then the game is a tie.
List to start:
2, 4, 10, 4, 1, 10
2, 4, 2, 5, 4, 1, 10
2, 4, 2, 5, 5, 10
2, 6, 5, 5, 10
2, 6, 5, 15
2, 6, 20
8, 4, 5
2, 3, 2, 5
Player 1 wins.