Disgaea had the largest numbers of any game I'd played before Cookie Clicker.
Cookie Clicker had the largest numbers of any game I'd played before Sandcastle Builder. In sandcastle builder numbers go up to ~10^308, the maximal double precision floating point number. After you reach that, you reach "infinite". Then there are new recourses which you can start building up.
The biggest number I encountered was in a Magic Puzzle we made up. It goes like this: select 20 cards, which you will divide in 2 decks of 10. 2 players play a game, and you make every decision there is to make and decide the outcome of everything that is up to fate (shuffling, coin flipping, ...). When the game ends, the number of lives of either player is your score.
Special restriction: When someone else comes along and he is given the same 20 cards and the same powers, there has to be a score he cannot reach.
We started out with scores like 100, then after a day 1000000, then 10^100, then 10^(10^100), then some numbers I just couldn't grasp anymore...