Number Choosing Contest.

Let H_n(a,b) be defined as such:



And:

Let S(a,b) =
H_n(a,b) where
n = Ackermann(S(a-1,b),S(a,b-1)))

I guess the following:

S(Graham's Number, Twelve Googol)

S(Largest Known Prime, Second Largest Known Prime)

S(The second number I guessed, The first number I guessed)

S(The number of board states in Chess regardless of legality or supply of pieces, The number of possible Go games)

292292292292

Edit: sorry